Generalized Analytic Automorphic Forms in Hypercomplex Spaces [1 ed.] 3764370599, 9783764370596

Table of contents :
Contents......Page 8
Introduction......Page 10
1.1 Hypercomplex numbers and Cli.ord Algebras......Page 17
1.2 Vahlen groups and arithmetic subgroups......Page 21
1.3 Di.erentiability, conformality and analyticity in hypercomplex spaces......Page 25
1.4 Basic theorems of Cli.ord analysis......Page 33
1.5 Orders of isolated a-points, an argument principle and Rouch´e’s theorem......Page 44
1.6 The generalized negative power functions......Page 51
2.1 Multiperiodic Mittag-Le.er series......Page 65
2.2 Some results on the zeroes of the generalized cotangent and tangent......Page 74
2.3 Liouville type theorems for generalized elliptic functions......Page 76
2.4 Series expansions, divisor sums and Dirichlet series......Page 81
2.5 The integer multiplication of the Cli.ord-analytic Eisenstein series......Page 88
2.6 Characterization theorems......Page 95
2.7 Lattices with hypercomplex multiplication......Page 99
2.8 Bergman kernels of rectangular domains......Page 113
2.9 Szego kernels of strip domains......Page 124
2.10 Boundary value problems on conformally .at cylinders and tori......Page 126
2.11 Order theory and argument principles on cylinders and tori......Page 129
3 Clifford-analytic Modular Forms......Page 133
3.1 Rotation and translation invariant Eisenstein series......Page 134
3.2 Clifford-analytic modular forms in one hypercomplex variable......Page 140
3.3 Clifford-analytic modular forms in two and several hypercomplex variables......Page 147
3.4 Some remarks on Cli.ord-analytic modular forms in real and complex Minkowski spaces......Page 160
3.5 Some Perspectives......Page 163
Bibliography......Page 167
List of Symbols......Page 179
Index......Page 181

Citation preview

Frontiers in Mathematics

Advisory Editorial Board Luigi Ambrosio (Scuola Normale Superiore, Pisa) Leonid Bunimovich (Georgia Institute of Technology, Atlanta) Benoît Perthame (Ecole Normale Supérieure, Paris) Gennady Samorodnitsky (Cornell University, Rhodes Hall) Igor Shparlinski (Macquarie University, New South Wales) Wolfgang Sprössig (TU Bergakademie Freiberg)

Birkhäuser Verlag Basel • Boston • Berlin

Author’s address: Rolf Sören Krausshar Department of Mathematical Analysis Ghent University Galglaan 2, Building S-22 9000 Ghent Belgium e-mail: [email protected] 2000 Mathematical Subject Classification 11F03, 11F55, 30G35, 11G15, 32A25, 53C27, 42B35 Library of Congress Cataloging-in-Publication Data Krausshar, Rolf Sören, 1972Generalized analytic automorphic forms in hypercomplex spaces / Rolf Sören Krausshar. p. cm. – (Frontiers in mathematics) Includes bibliographical references. ISBN 0-8176-7059-9 (acid-free paper) – ISBN 3-7643-7059-9 (acid-free paper) 1. Automorphic forms. 2. Hardy spaces. 3. Functions of complex variables. I. Title. II. Series Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de.

ISBN 3-7643-7059-9 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2004 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Cover design: Birgit Blohmann, Zürich, Switzerland Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN 3-7643-7059-9 987654321

www.birkhauser.ch

To Cristina, Francisca and my parents in love

Contents Introduction 1

Function theory in hypercomplex spaces 1.1 Hypercomplex numbers and Clifford Algebras . . . 1.2 Vahlen groups and arithmetic subgroups . . . . . . 1.3 Differentiability, conformality and analyticity in hypercomplex spaces . . . . . . . . . . . . . . . 1.4 Basic theorems of Clifford analysis . . . . . . . . . 1.5 Orders of isolated a-points, an argument principle and Rouch´e’s theorem . . . . . . . . . . . . . . . . 1.6 The generalized negative power functions . . . . .

ix

. . . . . . . . . . . . . . . . . .

1 1 5

. . . . . . . . . . . . . . . . . .

9 17

. . . . . . . . . . . . . . . . . .

28 35

2

Clifford-analytic Eisenstein series associated to translation groups 49 2.1 Multiperiodic Mittag-Leffler series . . . . . . . . . . . . . . . . . . 49 2.2 Some results on the zeroes of the generalized cotangent and tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.3 Liouville type theorems for generalized elliptic functions . . . . . . 60 2.4 Series expansions, divisor sums and Dirichlet series . . . . . . . . . 65 2.5 The integer multiplication of the Clifford-analytic Eisenstein series 72 2.6 Characterization theorems . . . . . . . . . . . . . . . . . . . . . . . 79 2.7 Lattices with hypercomplex multiplication . . . . . . . . . . . . . . 83 2.8 Bergman kernels of rectangular domains . . . . . . . . . . . . . . . 97 2.9 Szeg¨o kernels of strip domains . . . . . . . . . . . . . . . . . . . . . 108 2.10 Boundary value problems on conformally flat cylinders and tori . . 110 2.11 Order theory and argument principles on cylinders and tori . . . . 113

3

Clifford-analytic Modular Forms 3.1 Rotation and translation invariant Eisenstein series . . . . . . . . . 3.2 Clifford-analytic modular forms in one hypercomplex variable . . . 3.3 Clifford-analytic modular forms in two and several hypercomplex variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 118 124 131

viii

Contents 3.4 3.5

Some remarks on Clifford-analytic modular forms in real and complex Minkowski spaces . . . . . . . . . . . . . . . . . . . . . . . 144 Some Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Bibliography

151

List of Symbols

163

Index

165

Introduction Automorphic functions and automorphic forms are functions that show an invariance behavior or quasi-invariance behavior, respectively, under the action of a discrete group. For a number of reasons one is in particular interested in automorphic forms that are endowed with certain analytic properties. Examples of holomorphic automorphic forms and functions of one complex variable appeared first systematically in the paper [41] by G. Eisenstein (1847) and in lectures from K. Weierstraß in 1863. In [41] G. Eisenstein introduced the function series    1  1 1 n = 1,  z+ z+m − m m∈Z\{0} (1)  (1)
n (z; Z) := 1  n ≥ 2, (z+m)n m∈Z

and
(2) n (z; Ω)

  :=



1 z2

+



 

ω∈Ω\{0} ω∈Ω

1 (z+ω)2

−

1 ω2



n = 2, n ≥ 3,

1 (z+ω)n

(2)

where Ω denotes a two-dimensional lattice in C. These series are automorphic functions for a translation group generated by one or two generators. They are sometimes called meromorphic Eisenstein series for translation groups. They provide building blocks for the complex holomorphic trigonometric and elliptic functions in one complex variable. The Laurent expansion of the series
2 (z; Z + Zτ ) leads in turn to the function series  (cτ + d)−n Im(τ ) > 0 n ≥ 4, n ≡ 0 mod 2 (3) Gn (z) = (c,d)∈Z×Z\{(0,0)}

which represent holomorphic functions on the upper half-plane H + (C) := {z ∈ C | Im(z) > 0} that satisfy for all z ∈ H + (C) the transformation law  a b f (z) = (f |n M )(z), for all M = ∈ SL(2, Z), c d where (f |n M )(z) := (cz + d)−k f

(4)

az + b 

. cz + d The Eisenstein series Gn (z) provide thus examples of holomorphic automorphic forms for the full modular group SL(2, Z). A more systematic method to construct examples of holomorphic functions satisfying (4) is to start with a holomorphic function f˜ : H + (C) → C that is

x

Introduction

funcbounded on H + (C) and that belongs already to the class of automorphic  1 1 tions for the translation group T , the group generated by the matrix 0 1 which induces the translation z → z + 1. Summing the expressions (f˜|n M )(z) over a complete system of representatives of right cosets of SL(2, Z) modulo the translation invariance group T , i.e.,  (f˜|n M )(z), (5) M :T \SL(2,Z)

gives a convergent series for sufficiently large n whose limit function is then a modular form with respect to the full modular group SL(2, Z). These types of function series are often called Poincar´e series in a wider sense. The simplest non-trivial examples can be obtained by putting f˜ ≡ 1, thus yielding — up to a normalization factor — the classical Eisenstein series Gn (z). In turn the classical Eisenstein series Gn serve as building blocks for all modular forms of even positive entire weight n ≥ 4 for the full modular group SL(2, Z). However, the construction method proposed in (5) has a significant advantage: it can easily be adapted to the framework of other discrete groups, for instance, important congruence subgroups. The systematic development of the general theory of holomorphic automorphic forms of one complex variable is mainly due to H. Poincar´e, F. Klein and R. Fricke. See for instance [127, 81]. The theory of holomorphic automorphic forms and functions turned out to be a fundamental tool for solving many fundamental problems in number theory, including the determination of explicit relations between sums of divisors and representation numbers of quadratic forms, the construction of class fields, and even Fermat’s last theorem. Further areas of applications within mainstream mathematics include the study of Bergman and Hardy spaces of polyhedron type domains in C, the representation of Riemann surfaces and related partial differential equations and problems from index theory and algebraic geometry. Physicists use automorphic forms for instance in Yang–Mills theory and quantum gravity. To get a general idea about applications to physics, see for instance [67, 19] among many other contributions. Looking for higher dimensional analogues of the classical holomorphic automorphic forms of one complex variables is therefore strongly motivated under a lot of different viewpoints. There are several possibilities to extend classical function theory to higher dimensions. On the one hand there is the theory of several complex variables (see e.g., [76]) which is endowed with the ordinary regularity concept of complex holomorphy in each complex variable. Several complex variables theory has already become a well established branch inside mainstream mathematics since the 1930s and 1940s.

Introduction

xi

A further and different approach considers Clifford algebra-valued functions of one hypercomplex variable (see e.g. [13]) that are in the kernel of higher dimensional Cauchy–Riemann type operators. This function theory is today most often called Clifford analysis. Especially in the previous two decades Clifford analysis has emerged more and more as a valuable counterpart to the theory of several complex variables. It offers a number of generalizations of classical theorems from complex analysis, including a Cauchy integral formula, a residue theory, and even an argument principle. Already in the first half of the twentieth century higher dimensional (but complex-valued) generalizations of the classical automorphic forms from the plane have been developed within the framework of several complex variables theory. This line of investigation started with O. Blumenthal [9] in 1904 and continued with the school of C. L. Siegel and H. Maaß in the 1930s and early 1940s [153, 115]. One of the first authors to consider automorphic forms in a hypercomplex variable was R. Fueter. In 1927 R. Fueter constructed in his paper [54] automorphic forms and functions related to Picard’s group in a three-dimensional hypercomplex variable. However, the functions treated in [54] and also in his follow-up paper [55], are not in kernels of higher dimensional Cauchy–Riemann type operators. In 1949 H. Maaß proceeded to introduce in [117] another type of automorphic forms in a hypercomplex variable. The class of automorphic forms introduced by him consists of complex-valued non-analytic eigenfunctions to the higher dimensional hyperbolic Laplace–Beltrami operator. They are often called Maaß wave forms. They provide a natural generalization of their well-known two-dimensional non-analytic analogues in the complex upper half-plane which were introduced only a short time earlier by H. Maaß in [116]. All these works had a remarkable influence: Afterwards many authors extended these theories under numerous different aspects. Just to give a short overview over some of the most important mainstreams and recent developments, we recall that for instance, on the one hand, variants of holomorphic Siegel modular forms that are related to quaternionic symplectic groups or, more generally to orthogonal groups, have been studied, as for example by A. Krieg [99], by E. Freitag and C. Hermann [50] or very recently by e.g., F. B¨ uhler [17]. On the other hand for example E. K¨ ahler [80], J. Elstrodt, F. Grunewald, J. Mennicke [42, 45] as well as A. Krieg [100, 101] and V. Gritsenko [65] and others considered in several works complex-valued generalizations of the non-analytic automorphic forms with respect to discrete subgroups of the real Vahlen group in higher dimensional real half-spaces. Very recently O. Richter et al. for example considered also automorphic forms on m-fold Cartesian products of quaternionic half-spaces (see e.g., [131, 132]). Most of these generalizations defined in higher dimensional spaces are scalar valued. Vector-valued variants became of a growing interest, too (see e.g., [39, 15, 131]).

xii

Introduction

None of the generalizations of automorphic forms in a hypercomplex variable listed so far fit within the regularity concepts of Clifford analysis. They are not nullsolutions to higher dimensional Cauchy–Riemann type equations. In the period of 1998–2002, we were concerned to develop also a general theory of Cliffordvalued automorphic forms within the special framework of Clifford analysis and its regularity concepts. As far as we know, before 1998 research in this direction was primarily focussed on the construction of higher dimensional versions of the doubly periodic holomorphic Weierstraß functions, as for instance in [40, 61, 63, 67, 133, 68]. Additionally, in [128] a class of k-fold periodic hypercomplex-meromorphic functions on sector domains in IRk+1 has been considered — however, under rather different viewpoints and with different intentions. In [165] G. Z¨ oll made an attempt to construct monogenic functions that show an invariance behavior under discrete subgroups of Vahlen’s group that have the property of containing only finitely  a b many matrices with c = 0. This attempt however is very restrictive. c d Restricting to matrix groups with this property, rules out all hypercomplex generalizations of the classical modular group immediately. However, G. Z¨ oll’s aim was primarily to give one example that is different from the generalized Weierstraß elliptic functions. Unfortunately, the only concrete example that he gave then explictly turned out to coincide with the zero function. Nevertheless, this attempt provided a motivation for doing a deeper and more systematic research in this direction. In the previous five years, we started then to fill this gap step by step and to develop systematically a theory of monogenic and polymonogenic hypercomplexvalued automorphic forms in the setting of Clifford analysis in real Euclidean spaces and in arbitrary real and complex Minkowski types spaces. In [84, 85, 87, 92] for example we studied intensively function theoretical and number theoretical properties of monogenic and s-monogenic vector-valued Eisenstein series in IRk+1 with s ≤ k that are associated to arbitrary discrete translation groups. On the one hand we hereby extended the theory of generalized elliptic functions under several viewpoints. On the other hand we introduced in IRk+1 , to any arbitrary p-dimensional lattice in IRk+1 with 1 ≤ p ≤ k, p-fold periodic monogenic and polymonogenic generalizations of several classical trigonometric functions, including the tangent, the cotangent, the secant and the cosecant function. The results provide moreover a counterpart and a complement to the recent papers [105, 106] in which analogues of the Weierstraß ζ and ℘-functions were constructed in the context of the Fueter–Sce equation D∆m f = 0 in the particular space IR⊕ IR2m+1 . We showed that the s-monogenic generalized trigonometric and Weierstraß functions have many characteristic properties in common with the classical ones. In particular, they satisfy similar characteristic duplication formulas. More generally, we developed general integer multiplication formulas for s-monogenic Eisenstein series related to arbitrary translation groups in IRk+1 . We proved that a

Introduction

xiii

non-constant polymonogenic function that satisfies one of these specific integer multiplication formulas of the s-monogenic Eisenstein series must have singularities. Furthermore, it coincides, at least up to a constant, with one of those polymonogenic Eisenstein series whenever the singularities are all distributed in the form of a lattice and have all the same order and principal parts. Moreover, we studied the hypercomplex multiplication of the s-monogenic translative Eisenstein series in arbitrary finite space dimensions and derived explicit formulas for the trace of their hypercomplex division points. The transfer of the concept of complex multiplication to the setting of Clifford-analytic multiperiodic functions focuses on paravector-valued lattices whose paravector components stem from multi-quadratic number fields and are characterized in terms of special integral conditions. This in turn links special functions from Clifford analysis with particular algebraic number fields. In our joint paper with T. Hempfling [74] some results on isolated zeroes were developed for this function class. In particular we gave a very first insight in the fundamental value distribution of isolated a-points and showed for some special cases that the number of isolated poles and isolated a-points, counting multiplicity of the generalized monogenic elliptic functions, stands in a close relationship with each other. One thus obtains at least under particular conditions a certain analogue of the third Liouville theorem. The main tool for this analysis is the generalized argument principle for isolated a-points of monogenic paravector-valued functions that was developed before in [74]. In joint works with D. Constales [25, 24] and J. Ryan [96] applications of the Clifford-analytic translative Eisenstein series to function spaces, spin geometry and partial differential equations arising in Clifford and harmonic analysis were developed. To go a bit more into detail, in [25] it has been shown that the p-fold periodic monogenic Eisenstein series (1 ≤ p ≤ k + 1) of the pole order k + 1 are the building blocks for the reproducing kernel functions of the space of monogenic functions that are square-integrable over a rectangular domain in IRk+1 that is bounded in k + 1 − p directions. In [24] an explicit formula of the reproducing kernel function of the space of monogenic functions in a strip domain IRk+1 which have L2 -boundary values is derived in terms of the monogenic cosecant function. In [96] s-monogenic Eisenstein series of the pole order k are used to construct Cauchy and Green kernels on conformally flat cylinders and tori which arise from factoring out IRk by the translation group. They in turn admit the study of a number of classical boundary value problems on these manifolds, including the Dirichlet problem. Further to this, they lead to the development of useful techniques for the study of Hardy spaces that arise in this context, including explicit Plemelj projection formulas and Kerzman–Stein formulas. In [93] we developed explicit argument principles and a Rouch´e type theorem for monogenic functions with isolated a-points that are defined on the surface of such a cylinder or torus. In [84, 86] we used the monogenic translative Eisenstein series to introduce kfold periodic generalizations of the Eisenstein series (3) to Clifford analysis. Their

xiv

Introduction

Fourier expansions involve explicit representation numbers of sums of divisors and a vector-valued variant of the Riemann zeta function which is built with monogenic functions and defined on arbitrary finite-dimensional lattices in IRk+1 . In [23] a detailed study on the structure of this variant of the Riemann zeta function was provided and explicit formulas in terms of Dirichlet type series with polynomial coefficients were established. Returning to the monogenic generalizations of the Eisenstein series (3) that we defined in [84, 86], we observed that these series have several number theoretical properties in common with their classical counterparts from complex analysis, as for instance the structure of the Fourier expansion. However, in contrast to the classical complex case, only their set of singularities shows an invariance behavior under the inversion. On the half-space these generalizations are not quasi-invariant under inversions. In [88, 89, 90, 91, 93] we finally managed to construct step by step also s-monogenic automorphic forms that show a quasi-invariance under larger arithmetic subgroups of the Vahlen group, in particular under hypercomplex generalizations of the modular group and their principal congruence subgroups. To this end, we could once more use special s-monogenic translative Eisenstein series, in particular the generalizations of the series (3) introduced before in [84, 86], as generating functions in Poincar´e type series which provide us with non-trivial Clifford and vector-valued s-monogenic automorphic forms for larger arithmetic subgroups. In [90] we also introduced Clifford-valued s-monogenic Hilbert modular forms for products of hypercomplex arithmetic subgroups acting on Cartesian products of higher dimensional hyperbolic spaces. All the constructions can be adapted to the context of arbitrary real and complex Minkowski type spaces. These results open systematically the door to extend the applications of the s-monogenic translative Eisenstein series to function spaces, boundary value problems and spin geometry to a more general context. Partial results in this sense could be obtained in [27] and [97]. This book gives a comprehensive summary on the research results in this field developed over the last five years, either published only in form of articles or not yet published at all. As indicated in the particular text passages, some parts of the book contain results from joint articles with D. Constales, R. Delanghe, T. Hempfling, H. Malonek, and J. Ryan. The text is composed into three chapters. The first chapter deals with results of general function theoretic interest. We introduce the most important notions and explain the basic context. Then we treat fundamental questions related to concepts of differentiability, conformality and analyticity in the context of Clifford analysis. After this we recall basic theorems from Clifford analysis. Then we turn to fundamental questions from order theory and

Introduction

xv

to the development of a higher dimensional analogue of an argument principle. Next we give some useful representation formulas on elementary s-monogenic rational functions of negative homogeneity degree providing the building blocks of s-monogenic automorphic Eisenstein and Poincar´e series. Chapter 2 deals with s-monogenic Eisenstein series associated to translation groups and their applications where we treat in the topics mentioned here in passing. In Chapter 3 we treat s-monogenic automorphic forms for more general polyhedron type arithmetic subgroups of Vahlen groups and to their principal congruence subgroups. This book is based on my Habilitationsschrift [94] which I have prepared in the period October 2000–May 2003 at the Department of Mathematical Analysis of Ghent University (Belgium) and defended at the Technical University of Aachen (Germany) in an extremely friendly atmosphere at both places. To both the members of the Department of Mathematical Analysis (Ghent University) as of the Faculty of Mathematics, Natural Sciences and Computer Sciences (RWTH Aachen) and the further exterior referees of this work I am therefore deeply indebted. In particular I wish to express a special gratitude to Denis Constales for the numerous valuable discussions on a number of parts of this book, to Alan Offer for all his help on specific linguistical and particular typsetting questions as well as to the staff of Birkh¨auser Verlag, in particular to Thomas Hempfling, for expert advise in editing and publishing. Last but not least I wish to thank very much all my family and friends, especially my wife Cristina and my daughter Francisca, for all their love, patience, understanding and support in any concern during all the time. Rolf S¨ oren Kraußhar, Ghent, December 2003

Chapter 1

Function theory in hypercomplex spaces 1.1

Hypercomplex numbers and Clifford Algebras

Hypercomplex numbers are, roughly speaking, generalizations of the complex numbers in the sense of having several imaginary units. While a complex number represents a two-dimensional vector from IR2 , a hypercomplex number represents a vector in spaces of higher dimensions. The treatment of vectors from IR2 in terms of complex numbers has the additional advantage that a multiplication operation can be defined on IR2 . The systematic development of hypercomplex number systems can be traced back until the first half of the 19th century. A rather detailed historical survey about the first steps can be found for instance in the first part of [4]. In the first half of the 19th century, R. W. Hamilton looked for a threedimensional analogue of the complex numbers for a rather long time. However, a three dimensional analogue of the complex number field with the structure of a division algebra does not exist. To meet these ends one has to work at least in four dimensions. In this sense, R. W. Hamilton introduced in 1843 the quaternions IH which are numbers of the form q = q0 + iq1 + jq2 + kq3 where the imaginary units satisfy i2 = j 2 = k 2 = −1 and ij = k = −ji, jk = i = −kj, ki = j = −ik. In contrast to the complex number field the quaternions are not commutative and form a skew field. J. T. Graves and A. Cayley invented independently in 1843 and 1844, respectively, the eight-dimensional octonions O which are also often called the Cayley numbers, today. The octonions are numbers that consist of a real part and seven imaginary parts. They have the form z = x0 i0 + x1 i1 + · · · + x7 i7 , where i1 , i2 are the quaternionic units i, j, respectively, i3 = w is a further independent unit,

2

Chapter 1. Function theory in hypercomplex spaces

i4 = k, i5 = iw, i6 = jw and i7 = kw. The octonions are closed under multiplication. However, they are both non-commutative and non-associative. All the imaginary units il with l ∈ {1, . . . , 7} satisfy i2l = −1, i0 il = il i0 = il and il im = −im il for all m ∈ {1, . . . , 7} with m = l. The multiplication in O is explained completely by the following multiplication table of the imaginary units: · i1 i2 i3 i4 i5 i6 i7

i1 −1 −i4 −i5 i2 i3 i7 −i6

i2 i4 −1 −i6 −i1 −i7 i3 i5

i3 i5 i6 −1 i7 −i1 −i2 −i4

i4 −i2 i1 −i7 −1 i6 −i5 i3

i5 −i3 i7 i1 −i6 −1 i4 −i2

i6 −i7 −i3 i2 i5 −i4 −1 i1

i7 i6 −i5 i4 −i3 i2 −i1 −1

The multiplication of the element il with im is the element in the l-th row and the m-th column (not counting the first row above the horizontal line and the first column left of the vertical line). For the basic properties of octonions, see for example [4]. Due to the famous theorem of Frobenius, respectively Hurwitz [51], the algebras IR, C and IH are (up to isomorphy) the only real associative division algebras, while IR, C, IH, O are the only normed real algebras. In the following period the development of hypercomplex numbers continued mainly in two directions. One approach aimed in the direction of generalizing the quaternions by hypercomplex numbers that form associative algebras which culminated in the invention of the Clifford numbers by W. K. Clifford in 1879. A second approach was based on the so-called Cayley–Dickson construction which leads to 2k -dimensional both non-commutative and non-associative algebras extending the octonions. In this work we will mainly focus on the treatment of higher dimensional spaces by using Clifford algebras. For our needs we recall briefly the most important notions about Clifford algebras over general real and complex vector spaces. For more details we refer the reader for example to [13] and [37]. Let IK stand for the field of real numbers IR or complex numbers C. Let k be a positive integer, and let p, q be non-negative integers with p + q = k. In what follows we consider a k-dimensional vector space over IK that is endowed with a non-degenerate quadratic form Q of signature (p, q). The attached bilinear form B is then given by B(x, y) :=

 1

Q(x + y) − Q(x) − Q(y) 2

x, y ∈ IKk .

Let e1 , . . . , ek be the standard basis in the associated quadratic space IKp,q so that Q(e1 ) = · · · = Q(ep ) = −1, Q(ep+1 ) = · · · = Q(ek ) = 1 and B(ei , ej ) = 0 for i = j.

1.1. Hypercomplex numbers and Clifford Algebras

3

The attached Clifford algebra Clp,q (IK) is then the free algebra that is generated by IKk modulo the relation x2 = −Q(x)e0

x ∈ IKk

(1.1)

where e0 stands for the neutral element in the Clifford algebra. The relation (1.1) involves the following multiplication rules for the basis elements of the underlying vector space: e2i = 1 for i = 1, . . . , p, e2i = −1 for i = p + 1, . . . , k and ei ej + ej ei = 2δij e2i . A IK-basis for Clp,q (IK) is given by the set {eA : A ⊆ {1, . . . , k}} with eA = el1 el2 · · · elr , where 1 ≤ l1 < . . . < lr≤ k, e∅ = e0 = 1. The scalar and the vector part of an arbitrary element a = A⊆{1,...,k} aA eA where aA ∈ IK will be denoted by Sc(a) and by V ec(a). A Clifford number from Clp,q (IK) that has only a scalar and a vector part is called a paravector. The set of paravectors in Clp,q (IK) will be denoted by Ak+1 (IKp,q ). Every paravector from Ak+1 (IKp,q ) will be represented in the form z = x0 + x where the vector part will be denoted by a bold face letter. In the particular case IK = IR and p = 0, we simply write Ak+1 for the associated space of paravectors. Each element a ∈ Clp,q (C) can be written in the form a = a0 + ia1 with real Clifford numbers a0 , a1 from Clp,q (IR) where i denotes the complex imaginary unit. We write further Re(a) = a0 and Im(a) = a1 . One can subdivide the Clifford algebra Clp,q (IK) into an even and an odd part. The even part Cl+ p,q (IK) consists of Clifford numbers from Clp,q (IK) that can be represented in the form  aA eA a= A,|A|≡0 mod 2

where aA are elements from the underlying field IK and where |A| stands for the + cardinality of A. The odd part Cl+ p,q (IK) is defined analogously. Clp,q (IK) is a k−1 . subalgebra of Clp,q (IK) and its dimension over IK is 2 The complex number field is realized by C ∼ = Cl01 (IR) and is (up to isomorphy) the only non-trivial commutative Clifford algebra over IR. The subalgebra Cl+ 03 (IR) is isomorphic to the Hamiltonian skew field IH in the sense of an algebra isomorphism. One identifies the bivector units of Cl+ 03 (IR) with the quaternionic units i := e12 , j := e13 and k := e23 . The complexification Cl+ 03 (C) realizes then similarly the complex quaternions which were probably first used by W. K. Clifford in the period of 1873–1876.  The “Clifford” conjugation in Clp,q (IK) is defined by a = A aA eA where eA = elr el−1 · · · el1 ,

ej = −ej for j = 1, . . . , k and e0 = e0 = 1.

4

Chapter 1. Function theory in hypercomplex spaces

 |A|(|A|−1)/2 aA eA . Furthermore, the main The reversion is defined by a∗ := A (−1)  ∗  |A| involution is defined by a := A (−1) aA eA and one has a = a = a∗  . These (anti) automorphisms act on the basis elements eA ; they leave the complex unit i invariant. The “complex” conjugation, mapping an element a = a0 + ia1 from Clp,q (C) onto a0 − ia1 , shall be denoted by a in what follows. The Euclidean (Hermitian) scalar product in IKp,q is defined by z, w =

p 

k 

xj wj −

j=1

xj wj .

j=p+1 

It extends to Clp,q (IK) by a, b = Sc(ab ) and induces a pseudo semi-norm

a = |Sc(aa )| a ∈ Cl0k (IK) which is commonly called the Clifford norm. However, if a, b are arbitrary elements from Cl0k (IK), then we have only ab ≤ 2k/2 a

b in general. Let us turn again to the subspace of paravectors. In Clp,q (IK) a paravector z ∈ Ak+1 (IKp,q ) is invertible if and only if the expression N (z) := zz = x20 −

p  i=1

x2i +

k 

x2i

i=p+1

does not vanish. In this case z −1 = z/N (z). The set S0 := {z ∈ Ak+1 (IKp,q ), N (z) = 0} has in general the geometrical structure of a cone. The case IK = IR with p = 0 is a special case. In this case N (z) = x20 + · · · + x2k = z 2 for a paravector in Ak+1 , so that the null cone S0 in Ak+1 reduces to a single isolated point. Hence every non-zero paravector in Ak+1 is invertible. Let us further use the notation Sz˜ := {z ∈ Ak+1 (IKp,q ), N (z − z˜) = 0} for the singularity cone with center z˜. Here again Sz˜ reduces to the single point z˜ in the particular context of working in real Euclidean spaces. Every paravector z ∈ Ak+1 (IKp,q ) satisfies a quadratic equation of the form z 2 − S(z)z + N (z) = 0 where S(z) = 2x0 stands for the trace of the paravector.

1.2. Vahlen groups and arithmetic subgroups

5

In what follows we will mainly concentrate on working in Clifford algebras over real Euclidean spaces with p = 0, since they play a special role for the reasons that we pointed out. Generalizations to the more general setting of Clifford algebras over Minkowski type spaces are pointed out in some particular passages. In this text the symbol Z stands for the set of integers, IN stands for the set of positive integers and IN0 for that of non-negative integers. We will use the and standard multi-index notation. For multi-indices n = (n0 , n1 , . . . , nk ) ∈ INk+1 0 and paravectors z ∈ Ak+1 (IKp,q ) we write as usual j := (j0 , j1 , . . . , jk ) ∈ INk+1 0 z n := xn0 0 · · · xnk k , n! := n0 ! · · · nk !, |n| := n0 + · · · + nk ,    nk n n0 ··· , j ≤ n :⇔ j0 ≤ n0 , . . . , jk ≤ nk . := j j0 jk By τ (j) we further denote the multi-index n = (n0 , n1 , . . . , nk ) for which ni = δij , δij standing for the Kronecker symbol. We also write (a)p for the Pochhammer symbol a(a + 1) · · · (a + p − 1). Following for instance [120], the permutational product of arbitrary Clifford numbers a1 , . . . , an is defined by a1 × a2 × · · · × an :=

1 n!



ai1 · ai2 · · · · · ain .

perm(i1 ,...,in )

Let us further use the abbreviation a × · · · × a1 × · · · × an × · · · × an = [a1 ]k1 × [a2 ]k2 × · · · × [an ]kn = [a]k .

1  

  k1 times

kn times

Notice that the permutational product of Clifford numbers is commutative but not associative. In order to distinguish powers in terms of the permutational product from powers in the usual sense, we set round brackets whenever we mean ordinary powers. In this sense we write for example [a1 ]2 ×a2 = a1 ×a1 ×a2 , but (a1 )2 ×a2 = (a1 · a1 ) × a2 .

1.2

Vahlen groups and arithmetic subgroups

In this section we briefly recall the basic definitions of the higher dimensional analogues of the classical groups GL(2, C), GL(2, R), SL(2, R) and its arithmetic subgroups in the setting of using Clifford algebra-valued matrices that act on domains of IRk or Ak+1 , respectively. Approaches in this direction can be traced back to the beginning of the twentieth century. In 1902, K. Th. Vahlen discovered in [161] that one can describe M¨ obius transformations in higher dimensional

6

Chapter 1. Function theory in hypercomplex spaces

Euclidean spaces in terms of special (2 × 2) Clifford matrices. Unfortunately his ideas were forgotten for a rather long time. H. Maaß rediscovered K. Th. Vahlen’s approach in 1949 (cf. [117]). L. V. Ahlfors provided in the first half of the 1980s several more extensive contributions to the description of M¨ obius transformations by Clifford numbers and their geometrical mapping properties. His papers (see for instance [1]) contributed to a remarkable renaissance of K. Th. Vahlen’s ideas to a broader community. Many authors started to use more extensively Vahlen matrices in hyperbolic geometry, number theory and Clifford analysis. Just to point out some of the numerous contributions that followed shortly, see for instance [43, 44, 45, 165, 138] among many other important related works. Following the above mentioned references, we introduce Definition 1.1 (Clifford group). The Clifford group (C, IRk ) resp. (C, Ak+1 ) is defined as the set of elements a ∈ Cl0k (IR) which can be written in terms of a finite product of non-zero vectors from IRk , or paravectors from Ak+1 , respectively. Remark. In the case of working in arbitrary Minkowski type spaces, one considers instead of products of non-zero vectors or non-zero paravectors, vectors or paravectors that satisfy Q(x) = 0 or N (z) = 0, respectively. See for instance [141] and [29]. Next we recall the definition of Vahlen groups. Following [29] we introduce  a b Definition 1.2 (General Vahlen group). A matrix M = with coefficients c d k a, b, c, d out of (C, IR )∪{0} or from (C, Ak+1 )∪{0}, belongs to the general Vahlen group GV (IRk ) or GV (Ak+1 ), respectively, if additionally ad∗ − bc∗ ∈ IR\{0}, −1

a

−1

b, c

d ∈ IR

k

(1.2)

resp. ∈ Ak+1 , if c = 0 or a = 0, respectively. (1.3)

The general Vahlen group generalizes the group GL(2, C) to higher dimensions and permits a similar representation for M¨ obius transformations in IRk or Ak+1 as in the complex case. We recall Theorem 1.3. A function T : IRk ∪{∞} → IRk ∪{∞} is a M¨ obius transformation if and only if it can be written in the form T (x) = (ax+b)(cx+d)−1 with coefficients a, b, c, d ∈ (C, IRk ) that satisfy (1.2) and (1.3). obius transformation if A function T : Ak+1 ∪ {∞} → Ak+1 ∪ {∞} is a M¨ and only if it can be written in the form T (z) = (az + b)(cz + d)−1 with coefficients a, b, c, d ∈ (C, Ak+1 ) that satisfy (1.2) and (1.3). In view of z ∗ = z for all paravectors from Ak+1 (in particular, also for all vectors from IRk ) one can write T (z) = (az +b)(cz +d)−1 = (zc∗ +d∗ )−1 (za∗ +b∗ ). In the setting of arbitrary real and arbitrary real and Minkowski type spaces, (1.3) is replaced by ac∗ , cd∗ , db∗ , bc∗ ∈ IKp,q , permitting an analogous representation of

1.2. Vahlen groups and arithmetic subgroups

7

M¨obius transformations in arbitrary real and complex Minkowski type spaces, as mentioned e.g., in [141, 29]. Next we need: Definition 1.4 (Special Vahlen group). The special Vahlen group of SV (IRk ) or SV (Ak+1 ) is the subgroup of matrices belonging to GV (IRk ) or GV (Ak+1 ), respectively, whose coefficients satisfy moreover ad∗ − bc∗ = 1. The special Vahlen group can be regarded as a generalization of SL(2, C). As shown for instance in [43] the group SV (IRk ) or SV (Ak+1 ) is generated by the inversion matrix  0 −1 J := 1 0 and by translation matrices of the form  1 Ta := 0

a 1



where a ∈ IRk or a ∈ Ak+1 , respectively. The subgroup SV (IRk−1 ) acts transitively on the upper half-space of IRk , H + (IRk ) = {x ∈ IRk | xk > 0} by SV (IRk−1 ) × H + (IRk ) → H + (IRk ), x → M x := (ax + b)(cx + d)−1 . Consequently, all its subgroups leave the upper half-space invariant. The subgroup SV (Ak ) acts transitively on the upper half-space of Ak+1 , H + (Ak+1 ) = {z ∈ Ak+1 | xk > 0} by SV (Ak ) × H + (Ak+1 ) → H + (Ak+1 ), z → M z := (az + b)(cz + d)−1 . In view of the particular structure of the paravector space, it is sometimes advantageous to work instead on the right half-space of Ak+1 : H r (Ak+1 ) = {z ∈ Ak+1 | x0 > 0}. The appropriate analogue of SV (Ak ) for the right half-space is the modified special Vahlen group M SV (Ak+1 ) (see also [99, 100, 101] for the quaternionic case) which is generated by the modified inversion matrix  0 1 Q := 1 0

8

Chapter 1. Function theory in hypercomplex spaces

and by translation matrices of the form Ta where a ∈ IRk . Notice that M SV (Ak+1 ) is not a subgroup of SV (Ak+1 ), but a special subgroup of GV (Ak+1 ), since all matrices of M SV (Ak+1 ) have the property that their coefficients a, b, c, d satisfy either ad∗ − bc∗ = 1 or ad∗ − bc∗ = −1. Arithmetic subgroups of Vahlen groups that act discontinuously on the upper half-space were for instance considered in [44, 45]. We first recall the definition of the rational Vahlen group in V = IRk or V = Ak+1 , respectively. In what follows the set of rational numbers will be denoted as usual by Q. Definition 1.5.  (see e.g., [44, 45]) The rational Vahlen group SV (V, Q) is the set a b of matrices from M at(2, (C, V )) that satisfy c d (i) ad∗ − bc∗ = 1, (ii) ab∗ = ba∗ , cd∗ = dc∗ , (iii) aa, bb, cc, dd ∈ Q, (iv) ac, bd ∈ V , (v) axb + bx a, cxd + dx c ∈ Q (∀x ∈ V ), (vi) axd + bx c ∈ V

(∀x ∈ V ).

Next we need Definition 1.6. (cf. e.g., [44, 45]) A Z-order in a rational Clifford algebra is a subring R such that the additive group of R is finitely generated and contains a Q-basis of the Clifford algebra. The simplest examples of Z-orders in Cl0k (IRk ) are the standard Z-orders  Op := ZeA p ≤ k. A⊆P (1,...,p)

The following definition provides us with a number of arithmetic subgroups of the Vahlen group which act on the respective upper half-spaces. Definition 1.7. (cf. e.g., [44, 45]) Let I be a Z-order in Cl0k (IR) which is stable under the reversion and the main involution of Cl0k (IR). Then SV (V, I) := SV (V, Q) ∩ M at(2, I). For an n ∈ IN the principal congruence subgroup of SV (V, I) of level n is defined by    a b  SV (V, I; n) :=  a − 1, b, c, d − 1 ∈ nI . c d The following groups provide some special examples:

1.3. Differentiability, conformality and analyticity in hypercomplex spaces

9

Definition 1.8 (Special hypercomplex modular groups and their principal congruence subgroups). For p < k, resp. p < k + 1, let Γp (IRk ) := J, Te1 , . . . , Tep , resp. Γp (Ak+1 ) := J, Te1 , . . . , Tep . When itis clear which space IRk or Ak+1 is considered we simply write Γp . Let Op := A⊆P (1,...,p) ZeA be the standard order in Cl0p (IR). Then the associated principal congruence subgroups of Γp of level n are defined by    a b  Γp [n] := ∈ Γp  a − 1, b, c, d − 1 ∈ nOp . c d We have Γp [1] = Γp . All the groups Γp [n] are of course discrete and act all discontinuously on the upper half-space H + (IRk ) or H + (Ak+1 ), respectively. The element J has the order 4, i.e., J 4 = I, where I denotes the identity matrix. The group Tp . In elements T1 , . . . , Tp have infinite order and generate the translation  0 u∗ turn these matrices generate together with the matrices of the form 0 u−1 where u ∈ {±eA : A ⊆ {1, · · · , p}}, the subgroup 

  a b ∈ Γ Γ∞ := | c = 0 . (1.4) p p c d  1 b The elements from Tp with b ∈ nOp , n ∈ IN will be denoted by Tp [n] 0 1 in what follows. We further denote the translation group, associated to a general lattice Ωp = Zω1 + · · · + Zωp that is contained in the subspace IRk or Ak+1 , respectively, by T (Ωp ). Notice that the larger group generated by the matrices J, Tω1 , . . . , Tωp leaves the upper half-space H + (IRk ), resp. H + (Ak+1 ), invariant whenever Ωp ⊂ IRk−1 or Ωp ⊂ Ak respectively. However, this group does not act discontinuously on the respective half-space for all choices of generators of Ωp . Finally let us also introduce into a general p-dimensional lattice Ωp ⊂ Ak+1 the transversion group V (Ωp ) (cf. [66]) as   1 0 1 0  . ,··· , V (Ωp ) := ωp 1 ω1 1 Notice that the transversion group V (Ωp ) is conjugated with the translation group T (Ωp ). The transversion group associated with the orthonormal lattice will consequently be denoted by Vp .

1.3

Differentiability, conformality and analyticity in hypercomplex spaces

This section describes the transfer and the adaptation of the concept of classical complex holomorphy to functions of a hypercomplex variable. To this end let us

10

Chapter 1. Function theory in hypercomplex spaces

recall that the concept of complex holomorphy in the plane can be introduced in several equivalent ways. Among them there is the Cauchy approach which introduces holomorphy of a complex function f at a point z0 by the criterion of the existence of the limit of a complex differential quotient in an open neighborhood around z0 . Geometrically, non-constant holomorphic functions are precisely those functions that are locally orientation and angle preserving maps, i.e., locally conformal mappings in the sense of Gauss that preserve orientation. A third way is to introduce holomorphy by the criterion of the existence of a unique Taylor series representation of f around z0 . This approach is called the Weierstraß approach. Furthermore, there is the Riemann approach which provides an access to introduce holomorphic functions as real analytic functions that are in the kernel of the ∂ ∂ + i ∂y . Cauchy–Riemann operator D = ∂x G. Scheffers [147] was one of the first to consider quaternionic differential quotients of quaternion-valued functions of the form

 −1 (1.5) lim f (z + ∆z) − f (z) ∆z ∆z→0

or

lim

∆z→0

∆z

−1

 f (z + ∆z) − f (z)

(1.6)

at the end of the 19th century. As a consequence of the lack of commutativity this approach does not offer a rich function theory in quaternions. The set of quaternionic functions that satisfy (1.5) is restricted to linear affine functions of the form f (z) = az + b a, b ∈ IH while (1.6) implies that f (z) = za + b

a, b ∈ IH.

A complete proof for these statements was first provided by N. M. Krylov in 1947 (cf. [102]) and his student A. S. Melijhzon in 1948 (see [121]). The only functions for which both limits exist are functions of the form f (z) = αz + b where α ∈ IR and b ∈ IH. See also J.J. Buff’s paper [16] from 1970. A very elegant proof which uses the embedding of IH in C2 is given in [160]. The second approach, introducing hypercomplex-analyticity by conformality in the sense of Gauss, does not lead to a rich function theory, either. For convenience recall (cf. e.g. [7, 8]) that a real differentiable function f : G → IH, where G ⊆ IH is a domain, is called conformal in the sense of Gauss if there is a strictly positive real-valued continuous function λ : G → IR>0 , z → λ(z), such that

df 2 = λ(z) dz 2

1.3. Differentiability, conformality and analyticity in hypercomplex spaces

11

where 3  ∂f dxi df = ∂x i i=0

and

dz = dx0 + idx1 + jdx2 + kdx3 .

However, J. Liouville’s famous theorem tells us Theorem 1.9 (Liouville’s theorem). Let G ⊂ IRn be a domain. A C 1 function obius transformation f : G → IRn is a conformal map if and only if f is a M¨ whenever n ≥ 3. J. Liouville proved this theorem in 1850 (cf. [111]) under the assumption that f is a C 3 homeomorphism. More than one hundred years later, P. Hartman managed to prove this assertion in [69] for C 1 homeomorphisms. These proofs use essentially methods from differential geometry. Very recently a new and rather compact proof of this theorem, which uses primarily methods from Clifford analysis, has been given in [30]. In [83, 95] and in the recent overview paper [36] we have shown that one can extend the definition of the differential quotients (1.5) and (1.6) so that all M¨obius transformations and constant functions are included. This modification is based on the use of variable structural sets. A quaternionic structural set is a set of four quaternions that are mutually orthonormal. Structural sets are often used for example in works of M. Shapiro, N. Vasilevski and V.V. Kravchenko (cf. e.g., [98, 151]). What follows provides in particular an extension of [5]. For our needs it is crucial to use continuously variable structural sets, which may be interpreted geometrically as continuous moving frames. To be more precise: Definition 1.10. A variable continuous structural set in an open set U ⊆ IH is a set of four C 0 -functions Ψ0 (z), . . . , Ψ3 (z) that satisfy Ψi (z), Ψj (z) = δij at each z ∈ U. This tool in hand gives rise to the following notion of quaternionic differentiability in a more general sense which was introduced in [83, 95]. See also [36]. Definition 1.11. Let U ⊂ IH be an open set and let z˜ ∈ U with z˜ = x˜0 + ix˜1 + j x˜2 + k x˜3 . Then f : U → IH is called left quaternionic differentiable at z˜ if there exist three C 0 (U )-functions Ψ1 , Ψ2 , Ψ3 : U → spanIR {i, j, k} satisfying at each z˜ z ), Ψk (˜ z ) = δjk and such that the relation Ψj (˜ lim (∆z [Ψ] )−1 (∆f )

∆z [Ψ] →0

exists. Here, ∆z [Ψ] = ∆x0 +

3 

(1.7)

∆xi Ψi (˜ z ) with ∆xk = xk − x˜k .

i=1

f is called left quaternionic differentiable in U if f is left quaternionic differentiable at every point z ∈ U .

12

Chapter 1. Function theory in hypercomplex spaces

In an analogous way, right quaternionic differentiability is defined for f : U → IH, namely by requiring that for each z ∈ U , there exist three C 0 (U )-functions Φ1 , Φ2 , Φ3 : U → spanIR {i, j, k} such that Φj (z), Φk (z) = δjk and lim (∆f )(∆z [Φ] )−1

(1.8)

∆z [Φ] →0

exists. The limit lim (∆z [Ψ] )−1 (∆f )

∆z [Ψ] →0

may be considered as a linearization of the function f at the point z˜ with respect to the orthonormal basis [1, Ψ1 (˜ z ), Ψ2 (˜ z ), Ψ3 (˜ z )] and it is equal to the expression ∂f (˜ z ). It may be regarded as the left quaternionic derivative of f at the point z˜. ∂x0 One obtains the following nice analogy to the complex case: Theorem 1.12. Let G ⊂ IH be a domain. Then the set of left quaternionic differentiable functions in G coincides with the set of right quaternionic differentiable functions in G. A quaternionic differentiable function in G is either a conformal map in the sense of Gauss or a constant function in G. From Theorem 1.12 follows that one can associate with a quaternionic differentiable function two structural sets [Ψ] and [Φ] such that for k = 1, 2, 3, ∂f ∂f = Ψk (z) , ∂xk ∂x0

(1.9)

∂f ∂f = Φk (z). ∂xk ∂x0

(1.10)

and

In general, [Ψ] = [Φ], as a consequence of the non-commutativity. Proof of Theorem 1.12. If f is a constant function, then f is obviously both right and left quaternionic differentiable. Let us thus suppose that f : G → IH is a conformal map in G. According to the definition, there is a strictly positive continuous function λ : G → IR>0 such that

df 2 = λ(z)

3 

dx2k .

k=0

In view of df 2 = df df one hence obtains the relation 3  3

3 3     ∂fr 2  2 ∂fr ∂fr  dxi + 2 dxj dxi = λ(z) dx2k . ∂x ∂x ∂x i j i i=0 r=0 j 0 ∂x0 ∂x1 ∂x2 ∂x3 and

 ∂f ∂f  =0 i 0 centered around z˜. The symbol B(˜ z , R) stands for the closed ball around z˜ with radius R. Theorem 1.19 (Taylor expansion). Let f : B(˜ z , R) → Cl0k (IR) be left (right) monogenic. Then in each open ball B(0, r) with 0 < r < R, f (z) =

∞  |n|=0

Vn (z − z˜)an

or

f (z) =

∞ 

an Vn (z − z˜)

|n|=0



where the coefficients are uniquely given by a0 = f (˜ z ) and an = Here n stands for multi-indices of the form n = (0, n1 , . . . , nk ).

 ∂ |n| ∂xn f (z)

z=˜ z

.

The negative powers are replaced by the functions q0 (z) :=

z ∂ |n| ∂ n1 +···+nk and q (z) := q (z) = q0 (z), n ∈ INk+1 , |n| ≥ 1. n 0 0

z k+1 ∂xn ∂xn1 1 · · · ∂xnk k

The function q0 (z) coincides in the planar case k = 1 with the function z −1 . It is the fundamental solution to the D-operator and q0 (z − ζ) is the Cauchy kernel. The functions qn (z) with |n| ≥ 2 coincide in the case k = 1 up to a normalization constant with z −n for n ≥ 2. They turn out to provide the counterpart of the functions Vn in the Laurent expansion of monogenic functions with singularities. For convenience we recall:

20

Chapter 1. Function theory in hypercomplex spaces

Definition 1.20. A point z˜ ∈ Ak+1 is called a left (right) regular point of a left (right) monogenic function f , if there exists an ε > 0 such that f is left (right) monogenic in B(˜ z , ε). Otherwise, z˜ is called a singular point of f . z˜ is called an isolated left (right) singularity of f if it is a left (right) singular point of f for which one can find an ε > 0 such that f is left (right) monogenic in B(˜ z , ε)\{˜ z }. In analogy to classical function theory of one complex variable one can classify the singularities into three essentially different types, namely into removable singularities, unessential and essential singularities. However, in contrast to the classical case, a monogenic function in Ak+1 can have manifolds of singularities of dimension 0 (isolated singularities), and additionally of dimension 1, . . . , k−1. As a consequence of the Cauchy–Riemann equations, k-dimensional manifolds of singularities cannot appear. In order to recall the definition of essential and unessential singular sets of monogenic functions, which was first introduced for the quaternionic case by R. Fueter [61] and W. Nef [123, 124], we first need the following notions. Suppose that U ⊂ Ak+1 is an open set and that S ⊂ U is a closed subset. Let us consider around each s ∈ S a ball with radius ρ > 0 and let us denote by Hρ the hypersurface of the union of all balls centered at each s ∈ S with radius ρ. In the case where S is just a single isolated point, Hρ is simply the sphere centered at s with radius ρ. If S is a rectifiable line, or, more generally, a p-dimensional manifold with boundary, then Hρ is the surface of a tube domain. Let us denote by J(ρ) the limit inferior of the volumes of all closed orientable hypersurfaces HR with continuous normal field that contains S in the interior where we suppose that inf{ s − z˜ ; s ∈ S, z˜ ∈ HR } ≥ ρ. In terms of these notions one can now give the following definition: Definition 1.21. Let U ⊂ Ak+1 be an open set and let S ⊂ U be a closed subset. Suppose that f is left (right) monogenic in U \S and that f has singularities in each s ∈ S. Let Hρ ⊂ U be a hypersurface as defined above. The singular point s ∈ S is then called an unessential singularity of f if there is an r ∈ IN and an M > 0 such that z ) < M z˜ ∈ Hρ . (1.21)

ρr J(ρ)f (˜ In this case the minimum of the parameters r is called the order of the singular point s. If no finite r with this property can be found, then s is called an essential singularity. Functions that have at most unessential singularities and that are left (right) monogenic elsewhere are called left (right) meromorphic (in the sense of Fueter). In the case where S is an isolated point singularity, a rectifiable line or a manifold with boundary, one can substitute the condition (1.21) by z ) < M.

ρr f (˜

1.4. Basic theorems of Clifford analysis

21

Remarks. In classical complex function theory of functions in one complex variable any meromorphic function satisfies limz→˜z f (z) = ∞ at a pole z˜, independently on which path one approximates the singularity. In general, this is not true in higher dimensions (cf. [60, 129, 13, 136]). As a simple counterexample we note the function q1,1 (z) which has limit value zero when approaching the singularity 0 along the x0 -axis. In connection with this limit behavior J. Ryan has proved in [136] in the case k = 3, that provided f : A3 \{0} → Cl02 is a left monogenic function having a negative degree of homogeneity, then the set of lines radiating from the origin on which f vanishes, can at most have a finite cardinality. Let z˜ be an isolated singularity of f and let γ be an arbitrary path with γ(0) = z˜ and where f (γ(t)) is left (right) monogenic for all t > 0. For the quaternionic case, the Fueter school has provided examples where the set of limit values at a non-essential point singularity z˜ of a monogenic function f , i.e., {L ∈ IH ∪ {∞} |L = lim f (γ(t))} t→0

is a 2- or 3-dimensional manifold in IH ∪ {∞}. See for example [63] pp. 151–158. As a consequence of the Cauchy–Riemann system, 1-dimensional manifolds of limit values can never appear. This was proved in 1948 by Hs. Reich in [129]. Hs. Reich showed furthermore in his thesis [129] that this result remains even valid for the case where z˜ is a non-essential singularity lying on a two-dimensional manifold of non-essential singularities. For monogenic functions with isolated singularities one has the following kind of Laurent expansion theorem: Theorem 1.22 (Laurent expansion). Let f be left monogenic in the annular domain B(˜ z , R)\B(˜ z , r) ⊂ U with 0 < r < R. Then f has a Laurent series expansion of the form ∞ ∞   Vn (z − z˜)an + qn (z − z˜)bn , (1.22) f (z) = |n|=0

|n|=0

where both series converge normally in B(˜ z , R), resp. in Ak+1 \B(˜ z , r) and where n are multi-indices of the form (0, n1 , . . . , nk ). The coefficients an , bn are uniquely determined Clifford numbers and have the integral representation   1 1 qn (z) dσ(z)f (z), bn = Vn (z) dσ(z)f (z). an = Ak+1 Ak+1 ∂B(˜ z ,ρ)

∂B(˜ z ,ρ)

for an arbitrarily chosen real ρ within r < ρ < R. The coefficient b0 =: res(f ; z˜) is called the residue of f at z˜. Similarly to the complex case one can establish Theorem 1.23 (Residue Theorem). Suppose that U ⊂ Ak+1 is an open set. Let S ⊂ U be an oriented (k + 1)-dimensional compact differentiable manifold with

22

Chapter 1. Function theory in hypercomplex spaces

boundary. Assume that f : U → Cl0k (IR) is left monogenic in U with exception of a finite number of isolated points, say in y 1 , . . . , y r , such that y 1 , . . . , y r ∈ ∂S and that y 1 , . . . , y s ∈ int S where 1 ≤ s ≤ t. Then 

1

dσ(z)f (z) =

Ak+1

s 

res(f ; y i ).

i=1

∂S

An important tool for our needs is Theorem 1.24 (Mittag-Leffler theorem). Suppose that (ai )i is a sequence of distinct points in Ak+1 having no accumulation point in Ak+1 , so that we can associate to each point ai a non-negative integer pi and a function Qi (z) =

pi 

(i) qn (z − ai )bn ,

(1.23)

|n|=0 (i)

where the elements bn  are arbitrary Clifford numbers from Cl0k (IR). Then there is a function f : Ak+1 \ i {ai } → Cl0k (IR) which is left meromorphic in the whole space Ak+1 having poles of order (pi + k) at the points ai with the corresponding singular parts Qi , respectively. This function is uniquely defined up to a function that is left monogenic in the whole space. Remarks. W. Nef established in [123, 124] for the quaternionic case a more general version of the Laurent series expansion theorem and Mittag–Leffler theorem for functions f that have also sets of non-isolated singularities. These theorems in turn can also be generalized directly to arbitrary dimensions, as mentioned for instance in Section 1.6 of [84]. In these cases the expressions of the form qn (z − z˜)an in (1.22) and (1.23) are substituted more generally by  qn (z − c)d(an (c)),

(1.24)

S

involving the Lebesgue–Stieltjes integral over the singular set denoted by S. Also the residue theorem can be generalized to the more general context dealing with manifolds of singularities. See [61, 63] for the quaternionic case and [38] for the case in arbitrarily finite-dimensional real Euclidean spaces. In this case the residue of a function f at the singularity set S is said to be the first generalized Laurent coefficient in the sense of (1.24), i.e.  d(a0 (c)). Res(f ; S) = S

These kinds of residues are also known as Leray–Norguet residues (cf. [38]).

1.4. Basic theorems of Clifford analysis

23

Remark. All these theorems have complete analogues in the vector formalism in terms of the Dirac operator. In the vector formalism in IRk , the function q0 (z) is replaced by x q0 (x) := −

x k and the functions qm (z) are substituted by partial derivatives of q0 (x). In the vector formalism in IRk , one can restrict consideration to the partial differentiations in k − 1 directions, say for instance in the x1 , . . . , xk−1 directions. Similarly the analogues of the Fueter polynomials Vn (z) in the vector formalism are given by Vm1 ,...,mk−1 (x) :=

1  (xσ(i) + xk ek eσ(i) ) . . . (xσ(i) + xk ek eσ(i) ) m!

(1.25)

where m := m1 + · · · + mk−1 and σ(i) ∈ {1, . . . , k − 1} and the summation runs over all distinguishable permutations of the expressions (xσ(i) +xk ek eσ(i) ) without repetitions. One important property of the Dirac and Cauchy–Riemann operator in IRk and Ak+1 is that they factorize the Euclidean Laplace operator, viz DIRk = −∆k and DAk+1 DAk+1 = ∆k+1 , respectively. The monogenic functions are thus annihilated by the Euclidean Laplace operator, and every real component of a monogenic function is harmonic. More generally, let us consider Definition 1.25 (s-monogenic functions). Let U ⊂ IRk be an open set. Let s be a positive integer. Suppose that f : U → Cl0k (IR) is a C s -function. Then f is called left or right s-monogenic in U if DIsRk f = 0 or f DIsRk = 0, respectively. Sometimes we say for simplicity Clifford-analytic or polymonogenic for smonogenic. If s is even, then f ∈ Ker ∆s/2 . The analogue of the set of s-monogenic functions in the paravector formalism is the set of C s -functions in U ⊂ Ak+1 that satisfy D[s] f = 0, resp. f D[s] = 0, where we mean D[s] = ∆s/2 if s is an even positive integer and D[s] = D∆(s−1)/2 if s is an odd positive integer. We will call functions in Ker D[s] also s-monogenic and sometimes also simply Clifford-analytic. The analogues of the function q0 (x) in the framework of Ker Ds (vector formalism in IRk ) are given by  x  s odd with s ≤ k − 1,  k+1−s (s)

x (1.26) q0 (x) := 1   s even with s ≤ k − 1. k−s

x The s-monogenic analogues of qm (x) for |m| ≥ 1 are given in terms of (s) (x) := qm

∂ |m| (s) q (x). ∂xm 0

24

Chapter 1. Function theory in hypercomplex spaces

Whenever s > 1, we cannot restrict ourselves to only consider indices m with mk = 0. The analogues of the function q0 (z) in the framework of Ker D[s] (paravector formalism in Ak+1 ) are given by

(s) q0 (z)

(s)

q0

  

z

z k+2−s := 1  

z k−s (s)

s odd with s ≤ k, s even with s ≤ k.

and its partial derivatives qm (z) :=

∂ |m| (s) ∂z m q0 (z)

=

∂ n0 +···+nk (s) (z) n q n ∂x0 0 ···∂xk k 0

(1.27)

provide

canonical generalizations of the usual negative power functions in Ker D[s] from several different viewpoints. Again, whenever s > 1, one cannot restrict oneself exclusively to indices m with m0 = 0. Special attention shall be paid to the case where s = 2m + 1 and k = 2m + 2 in the context of the paravector spaces Ak+1 . The associated function class, first investigated by R. Fueter in 1931 [55] and M. Sce [146], is today often called the class of the Fueter–Sce solutions or the class of holomorphic Cliffordian 2m+2 (2m+1) (x0 + j=1 xj ej ) functions, see [104]. In this framework the functions q0 and its partial derivatives with respect to x0 coincide (up to a constant) with the ordinary negative powers of the hypercomplex variable z, see for example [104]. All 2m+2 positive and negative powers of the paravector variable z = x0 + j=1 xj ej are annihilated by this operator. It should be pointed out that the set of Fueter–Sce solutions contain the set of the hypermonogenic functions studied by H. Leutwiler et al. (see e.g., [107, 72, 47]), i.e., functions that are in the kernel of the hyperbolic Cauchy–Riemann operator Dhyp := xk D + (d − 1) as a special subset. The power functions of the paravector variable are also annihilated by the hyperbolic Cauchy-Riemann operator. The crucial point is that one can reconstruct locally every s-monogenic function from a finite number of 1-monogenic functions. One obtains a very simple representation in the vector formalism: Theorem 1.26 (Almansi type decomposition). (cf. e.g., [144]) Let B(0, R) ⊂ IRk be the open ball with radius R > 0 having its center at the origin. Let f : B(0, R) → Cl0k (IRk ) be a left s-monogenic function in B(0, R). Then there are s uniquely defined functions f1 , . . . , fs that are 1-monogenic in B(0, R) such that f (x) =

s  j=1

xj−1 fj (x).

1.4. Basic theorems of Clifford analysis

25

In the paravector formalism one gets a similar but slightly more complicated representation for a function in Ker D[s] of the form f (z) = f1 (z) + zf2 (z) + z 2 f3 (z) + z z 2 f4 (z) + · · · where the functions f1 , f2 , . . . are uniquely defined and satisfy Df2j−1 = 0, resp. Df2j = 0, where j is a positive integer with j < (s + 1)/2 if s is odd and j < s/2 is s is even. The Almansi decomposition leads to Green formulas for s-monogenic functions. To this end notice there are real constants C1 and C2 so that Dq0j+1 (z) = C1 q0j (z) and

(j)

∆q0j+2 (z) = C2 q0 (z) both for the paravector and the vector formalism. In order to get simple Green for(2) (3) (2) (3) mulas let us introduce normalizations of the functions q0 , q0 , . . . say q˜0 , q˜0 , . . . so that (2)

(3)

(4)

(5)

q0 (z) = Dq˜0 (z) = ∆˜ q0 (z) = D∆˜ q0 (z) = ∆2 q˜0 (z) = . . . and similarly for the vector formalism so that (j+1)

q0 (x) = Dj q0

(x)

for j < k. These functions serve as Green kernels for s-monogenic functions. In the vector formalism one obtains again very simple Green type formulas. Theorem 1.27 (Green type formulas for s-monogenic functions in IRk ). (cf. e.g. [144]) Let 0 < r < R and suppose that f : B(0, R) → Cl0k (IRk ) is left s-monogenic. Then   s−1 1 (j+1) ˜0 q (y − x)dσ(y)f (y). f (x) = Ak j=0 ∂B(0,r)

In the paravector formalism working in Ak+1 one gets a similar formula of the slightly more complicated form   (1) (2) Ak+1 f (z) = q0 (ζ − z)dσ(ζ)f (ζ) + q˜0 (ζ − z)dσ(ζ)Df (ζ) ∂B(0,r)



∂B(0,r) (3)



q˜0 (ζ − z)dσ(ζ)∆f (ζ) +

+ ∂B(0,r)

(4)

q˜0 (ζ − z)dσ(ζ)D∆f (ζ) + · · ·

∂B(0,r)

In particular one obtains a Green formula for the class of holomorphic Cliffordian functions which include the class of hypermonogenic functions (compare with [104, 47]).

26

Chapter 1. Function theory in hypercomplex spaces

All s-monogenic functions are totally invariant under the translation group. If f is s-monogenic in the variable z or x respectively, and if T is a matrix from T (Ωp ), then f (T z ) or f (T x ) remains s-monogenic. Moreover, the  composition a b of a left (right) s-monogenic function with a general matrix M = from c d k GV (IR ), resp. GV (Ak+1 ), remains left or right s-monogenic if one multiplies from (s) (s) the left with the factor q0 (cz + d) or from the right with the factor q0 (zc∗ + d∗ ), respectively. The following theorem gives a more precise formulation and provides a slight adaptation of Theorem 9 from [141] to the context of groups acting on the upper half-space H + (Ak+1 ).  a b Theorem 1.28. Let M = be a matrix from SV (IRk−1 ), resp. from c d SV (Ak ). Suppose f : H + (IRk ) → Cl0k (IRk ) is a solution to DIsRk f (x) = 0 or s analogously that g : H + (Ak+1 ) → Cl0k (IRk ) is a solution to DA g(z) = 0. Then k+1   (s) (1.28) DIlRk q0 (cx + d)f (M x ) = 0 for all l ≥ s for all x ∈ H + (IRk ) and all M ∈ SV (IRk−1 ). Similarly,   [l] (s) DAk+1 q0 (cz + d)g(M z ) = 0 for all

l ≥ s.

(1.29)

A similar result is obtained for the right s-monogenic case by instead multiplying (s) (s) the factor q0 (xc∗ + d∗ ) or q0 (zc∗ + d∗ ), respectively, from the right. Remark. Notice that z = z ∗ holds for all paravectors from Ak+1 . All vectors from IRk satisfy x = −x. However, the only paravectors that satisfy z = −z are vectors from IRk . This leads to an advantage in the vector formalism concerning the conformal invariance formula Theorem 1.28. If (c, d) is the second row from a Vahlen matrix SV (IRk ), then (cx + d)∗ = ±(cx + d) for each vector x ∈ IRk . Therefore we can substitute in the vector formalism (1.28) equivalently by  (s) DIlRk q0 (cx + d)∗ f (M x ) = 0

for all l ≥ s.

(1.30)

This causes a slight symmetry break which we will need later in Chapter 3.2–3.4 for the construction of non-vanishing s-monogenic Hilbert modular forms. In the following sections in Chapter 1, in Chapter 2 and Chapter 3.1 we will exclusively work in the paravector formalism. Everything can be developed analogously in the vector formalism in which one often gets simpler formulas. The (s) (s) main thing is to switch from the functions qm (z) to qm (x), etc.. In Chapter 3.2– 3.4 we prefer then to work in the vector formalism in order to make use of the

1.4. Basic theorems of Clifford analysis

27

mentioned symmetry break by interchanging the conjugation with the reversion in the conformal invariance formulae. Of course, everything that we develop in Chapter 3.2–3.4 can be developed similarly in the paravector formalism. However, in this case one needs to take instead of the reversion another anti-automorphism, [s] or to consider also anti s-monogenic functions, i.e., functions in Ker D . Remarks. A number of the central theorems presented in this section can be adapted quite nicely in a similar form to the more general setting of real and complex Minkowski type spaces. For details, see for instance [135, 134, 141, 29]. (s) In the real Minkowski type spaces Ak+1 (IRp,q ) the functions q0 (z) are replaced for many of those applications by the expressions  1  s ≡ 0 mod 2,  (k+1−s)/2 (s)

N (z) Q0 (z) = z   s ≡ 1 mod 2, (k+2−s)/2

N (z) (s)

(s)

(1)

and the functions qm by the partial derivatives of Q0 . The functions Qm are annihilated from the left and from the right by the associated Cauchy–Riemann operator in Ak+1 (IRpq ), i.e., DAk+1 (IRpq ) :=

p k   ∂ ∂ ∂ − ei + ej . ∂x0 i=1 ∂xi ∂xj j=p+1

(2)

The functions Qm are special nullsolutions to the wave operator ∆Ak+1

(IRp,q )

p k   ∂2 ∂2 ∂2 = − + . 2 2 ∂x0 i=1 ∂xi ∂x2j j=p+1 (s)

In the complexified case Z ∈ Ak+1 (IRp,q ) with k ≡ 1 mod 2 the analogues of qm are given by  1   s ≡ 0 mod 2,  (k+1−s)/2 [N (Z)] (s) Q0 (Z) = Z   s ≡ 1 mod 2  (k+2−s)/2 [N (Z)] (1)

and their partial derivatives. The functions Qm are annihilated from the left and (2) from the right by the complexified Cauchy–Riemann operator and Qm by the [s] complexified Laplace operator. Holomorphic functions in the kernel of DA (Cp,q ) k+1 are consequently called complex s-monogenic. In [135, 134, 139] complete analogies of the integral theorems given above (s) were established in terms of the functions Q0 (Z) in the setting of complexified Clifford analysis for the cases where k ≡ 1 mod 2. Notice that if both p and q differ

28

Chapter 1. Function theory in hypercomplex spaces (s)

from zero, then the functions Q0 do not only have a point singularity at zero. They become singular in the whole zero-quadric. In this context one therefore has to restrict oneself to the integration over special regions which are sometimes called real domain manifolds. Following the above cited papers, a real domain manifold M ⊂ Ak+1 (Cp,q ) is a smooth real (k + 1)-dimensional manifold, if for each Z ∈ M the tangent space T Mz is spanned by (k + 1) paravectors {cj,Z ej }kj=0 where each ˜ cj,Z ∈ C\{0} and where SZ˜ ∩ M = {Z}. In view of Theorem 10 from [141], one obtains analogies of Theorem 1.28 in special circular half-cones in any type of real or complex Minkowski space IKp,q (s) (s) in terms of the functions Q0 (z) and Q0 (Z), respectively. This will be explained (s) in a more detailed way later in Chapter 3.4. The use of the functions Qm (z) (s) and Qm (Z) admits furthermore a generalization of Mittag-Leffler’s theorem to the framework of arbitrary real and complex Minkowski-type spaces. The func(s) (s) tions Qm (z) and Qm (Z) provide thus important building blocks for s-monogenic functions in Ak+1 (IKp,q ) with prescribed singularity cones. However, in contrast to the Euclidean case, additional restrictions concerning the locations of the centers of the singularity cones have to be made in order to preserve convergence. In the complexified case for instance, the centers of the singularity cones are supposed to lie once more on IRk+1 -like domain manifolds. Since the singularity cones Sz˜ reduce always to a single point in the context of real Euclidean spaces, all the special additional restrictions disappear in Ak+1 . This reflects once more the special role of real Euclidean spaces to which we will return now again.

1.5

Orders of isolated a-points, an argument principle and Rouch´e’s theorem

Many classical theorems of classical complex analysis, in particular of value distribution theory, are established on order relations of a-points of holomorphic functions for whose quantitative description the argument principle plays the central role (cf. e.g., [78]). Classical complex analysis offers several approaches to define the order of an a-point. A lot of textbooks introduce the order of an a-point (a = ∞) of a holomorphic function f at a point c as the smallest non-negative integer for which one obtains a decomposition of the form f (z) − a = (z − c)k g(z)

(1.31)

with a holomorphic function g that has no zeroes within a sufficiently small chosen neighborhood around z = c. Alternatively, one can also introduce the order in the

1.5. Orders of isolated a-points, argument principle, Rouch´e’s theorem

29

following ways: ord(f − a; c)

:=

ord(f − a; c)

:=

ord(f − a; c)

:=

   dk  (f (z) − a) = 0 , min k ∈ IN0 ;  k dz z=c  1 1 dw, 2πi f (γ) w − a  1 f  (z) dz. 2πi γ f (z) − a

(1.32) (1.33) (1.34)

In (1.33) and (1.34) γ stands for a simple closed curve around c which has no other a-points in its interior and no a-points and poles on its path. (1.33) shows that the order of the a-point of f at c is the winding number of f ◦ γ counting how often the imagined curve f ◦ γ wraps around a. An important question is to analyze whether generalizations can also be developed in the Clifford analysis setting. The development of generalizations in this sense is far from being a straightforward extension from two to higher dimensions. On the one hand a monogenic function in Ak+1 needn’t have only isolated apoints. A non-constant 1-monogenic function may possess p-dimensional manifolds of a-points where 0 ≤ p ≤ k−1. Due to the Cauchy–Riemann system the case p = k appears only for constant functions. If s ≥ 2, then one can also find s-monogenic non-constant functions that have a k-dimensional manifold of a points. This is an important difference to the classical complex case. On the other hand it does not seem very sensible to introduce the notion of the order of an isolated a-point of a monogenic function simply by generalizing (1.31), since the multiplication of a polynomial with a general monogenic function does in general not give a monogenic function again. At this point it is natural to ask whether one can introduce the notion of the order of an isolated a-point of a monogenic function by generalizing (1.32). However, G. Z¨oll has shown in [165] that such an approach would in general not provide a notion of the order in the sense of the topological mapping degree, so described for instance in [2]. In this sense, G. Z¨oll gave the following example within the framework of quaternionic analysis (see for details [165], p. 131): f (z) = (x0 + ix1 )2 + (x2 − ix3 )3 j = −2V2,0,0 (z) + 6V0,3,0 (z)j + 6V0,0,3 (z)k − 6V0,2,1 (z)k − 6V0,1,2 (z)j. The function f (z) has an isolated zero point at the origin. It is exclusively composed by homogeneous terms of degree 2 and degree 3. However, following G. Z¨ oll, the topological mapping degree around zero is −6. In [59, 63] R. Fueter suggested defining the order of an isolated quaternionic a-point of a quaternionic monogenic function in the sense of generalizing the approach (1.33). This order definition describes then indeed the topological mapping degree of the function. In 2000, T. Hempfling extended this definition to the setting of Clifford analysis in arbitrary finite dimensional Euclidean spaces (cf. [73]).

30

Chapter 1. Function theory in hypercomplex spaces

This approach enables indeed a theoretical treatment of orders of isolated a-points of monogenic functions that take values in Ak+1 . However, the appearing integral in [73] is in general rather difficult to compute explictly, since one has to perform the integration over the image of the sphere under the given arbitrary function. A further important question that arose in that context was to ask whether it is possible to set up an explicit argument principle for monogenic functions in Ak+1 from this order definition. In the paper [74], jointly written with T. Hempfling, an explicit argument principle for isolated a-points of monogenic functions that take values in the paravector space Ak+1 has been set up for arbitrary dimensions. This argument principle in turn involves furthermore integrals which are easier to evaluate. It provides also a generalization of the particular reformulated order formula for quaternionic functions given in [59] p. 88 (Formula (5)) and in [63] on p. 199. The argument principle provided the basis for the treatment of a number of rather central questions concentrated around isolated a-points of monogenic functions. In particular, a generalized version of Rouch´e’s theorem could be developed in the context of isolated a-points of monogenic functions. Furthermore, it gave some first insight in the value distribution of monogenic generalizations of elliptic functions. For some particular cases we could show with this argument principle that there is an explicit balance relation between the a-points and the poles of the monogenic generalized elliptic functions. In this section we summarize some of the general results from [74]. The specific applications to generalized elliptic functions will be discussed in Chapter 2. In order to start we first recall the basic definition of isolated a-points of monogenic functions and the order definition (cf. [59, 63, 73]): Definition 1.29 (Isolated a-points). Let U ⊂ Ak+1 be an open set and f : U → Ak+1 be a function. Let a ∈ Ak+1 . Then f is said to have an isolated a-point at c ∈ U , if f (c) = a and if there is an ε > 0 such that f (z) = a for all z ∈ B(c, ε)\{c}. Remark. A point c ∈ U that satisfies limz→c f (z) = ∞, independently from the path, is called an ∞-point of f . As a consequence of the implicit function theorem one obtains Proposition 1.30 (Sufficient criterion for isolated a-points). (cf. [73]) Let f : U → Ak+1 be a local diffeomorphism. Let c ∈ U and f (c) = a. If the Jacobian determinant (det Jf )(c) = 0, then c is an isolated a-point of f . It shall be noticed that this criterion is again only a sufficient criterion. One can show more generally, that if rank(det Jf (c)) = k + 1 − p, then c lies on a p-dimensional manifold of a-points. See [73] for more details. This is a sufficient criterion, too. Within the framework of isolated a-points, T. Hempfling introduced in [73] the following definition of the order of an isolated zero point of a monogenic function that takes values in Ak+1 :

1.5. Orders of isolated a-points, argument principle, Rouch´e’s theorem

31

Definition 1.31. Let U ⊂ Ak+1 be a non-empty open set. Assume that f : U → Ak+1 is monogenic and that c ∈ U is an isolated zero of f . Let ε > 0 such that B(c, ε) ⊆ U and f |B(c,ε)\{c} = 0. Then the integer  1 q0 (y)dσ(y) (1.35) ord(f ; c) := Ak+1 f (∂B(c,ε))

is called the order of the zero point of f at c. The expression ord(f ; c) is actually an integer. The proof is an application of the generalized Cauchy’s integral formula which tells us that  1 q0 (z − y)dσ(z)g(z) = w∂G (y)g(y), Ak+1 ∂G for any function g that is left monogenic in a simply connected domain G ⊂ Ak+1 with sufficiently smooth boundary conditions. In particular, one obtains by putting g ≡ 1 and G = B(c, ε) that  1 q0 (z − y)dσ(z) = w∂B(c,ε) (y). (1.36) Ak+1 ∂B(c,ε) Next one substitutes in (1.36) y by f (c) and ∂B(c, ε) by f (∂B(c, ε)). Following for instance [2] p. 470, the image f (∂B(c, ε)) is actually again a k-dimensional cycle under the given conditions. This leads consequently to  1 q0 (z − f (c))dσ(z) = wf (∂B(c,ε)) (0). (1.37)

 Ak+1 f (∂B(c,ε)) =0

The expression wf (∂B(c,ε)) (0) counts how often the image of the sphere around the zero point wraps around zero and is hence an integer. It should be noticed for all that follows, that one may substitute more generally the sphere in formula (1.35) by a nullhomologous k-dimensional cycle parameterizing a k-dimensional surface of a (k + 1)-dimensional simply connected domain inside U that has c in its interior, no further zeroes in its interior and no zero on the proper cycle. This definition generalizes the classical definition of the order in the sense of (1.33). For the special quaternionic case it has already been formulated by R. Fueter in [59, 63]. For topological mapping reasons, it is important to claim that the range of values of f is contained in the paravector space Ak+1 when working in arbitrarily finite dimensional spaces. The quaternionic case turns out to play once more a very special role. In the case where f (c) = 0 one chooses ε sufficiently small so that f |B(c,ε) = 0. Then, (1.35) defines also the order of f at c. In this case ord(f ; c) = 0, which follows as a consequence from Cauchy’s integral theorem. Suppose that the function

32

Chapter 1. Function theory in hypercomplex spaces

g(z) = f (z) − a has an isolated zero at z = c. Then c is an isolated a-point of f . Substituting in (1.35) the function f by g = f − a gives then consequently the order of the isolated a-point of f at c. A contrast to the classical case consists of the fact that it is possible to have ord(f ; c) = 0 even if f (c) = 0 (see [63]). This phenomenon appears already in the quaternionic case. As mentioned previously one needs to perform in (1.35) the integration over the image of the sphere which is very difficult in most cases. To get simpler integrals which are easier to determine, one can apply the following transformation rule for differential forms proved by G. Z¨ oll ([165], §3.2): Lemma 1.32 (Transformation formula). Let G ⊂ Ak+1 be a domain and f : G → Ak+1 be continuously differentiable. Then dσ(f (z)) = [(Jf )∗ (z)] ∗ [dσ(z)]

(1.38)

where (Jf )∗ stands for the adjoint matrix of Jf . The multiplication ∗ denotes here the matrix-vector multiplication. In order to avoid confusion with the usual Clifford multiplication, we shall use the symbol ∗ when matrix-vector multiplication is meant and use brackets additionally. Proof. Following [165], let us write   

∂f ∗

∂f (z) := (−1)i+j det (z) (Jf )(z) = Adj ∂xj ∂xj i,j i,j

∗ ∂f where ∂x (z) denotes the matrix that is deduced from the Jacobi matrix Jf j by deleting the i-th row and the j-th column. One obtains: dσ(f (z))

=

=

k 

k  ∂(f0 , . . . , fi−1 , fi+1 , . . . , fk )  dxj (z) (−1) ei ∂(x 0 , . . . , xi−1 , xi+1 , . . . , xk ) i=0 j=0 i

k  k  i=0 j=0



   =  

Adj

=



∂f0 ∂x0 (z)

..

. Adj

∂f ∗ j (z) (z) (−1)j dx ∂xj

    ∂f0 · · · Adj ∂x (z) 0 (−1)0 dx k     .. .. .  .

.   k ∂fk (−1) dxk · · · Adj ∂xk (z)

(−1)i+j ei det



∂fk ∂x0 (z)

[(Jf )∗ (z)] ∗ [dσ(z)].




If one puts (1.38) for y = f (z) into (1.35), then one obtains the following formula for the order of an isolated a-point (a ∈ Ak+1 ) which is easier to evaluate:

1.5. Orders of isolated a-points, argument principle, Rouch´e’s theorem

33

Theorem 1.33. (cf. [74]) Let G ⊂ Ak+1 be a domain. Suppose that f : G → Ak+1 is monogenic in G and that c ∈ G is an isolated zero point of f . Let ε > 0 so that B(c, ε) ⊆ G and f |B(c,ε)\{c} = 0. Then ord(f ; c) =



1

    q0 (f (z)) (Jf )∗ (z) ∗ dσ(z) .

Ak+1

(1.39)

∂B(c,ε)

Formula (1.39) provides a generalization of the second quaternionic order formula given in [59] p. 88 (Formula (5)) and in [63] p. 199. An interesting effect that appears in formula (1.39) is that both matrix-vector multiplication and Clifford multiplication operations have to be performed. This effect did not appear that clear and explicit in R. Fueter’s quaternionic order formula, since he used quaternionic determinants instead. Let us also analyze how this formula generalizes the classical formula from complex analysis, i.e., ord(f ; c) =

1 2πi

 ∂B(c,ε)

f  (z) dz. f (z)

We observe that in the higher dimensional case the derivative f  (z) is generalized 1 is replaced by q0 (f (z)), by the adjoint matrix of the Jacobian. The expression f (z) which is rather natural since q0 (z) provides the monogenic generalization of the complex analytic function z1 . Next we may finally deduce the following argument principle for isolated a-points of monogenic functions. Theorem 1.34. Let G ⊂ Ak+1 be a domain and let f : G → Ak+1 be a monogenic function. Suppose that Γ is a nullhomologous k-dimensional cycle parameterizing a k-dimensional surface of a (k+1)-dimensional simply connected bounded domain E ⊂ G. If f has only isolated a-points in the interior of E and furthermore no a-points on ∂E, then  c∈E

ord(f − a; c) =

1



Ak+1

q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)].

(1.40)

Γ

Proof. Since E is a bounded domain, f can at most have a finite number of isolated a-points in E; we denote them by c1 , . . . , cm . Next take sufficiently small numbers ε1 , . . . , εm > 0 such that all the sets B˜1 := B(c1 , ε1 )\{c1 }, . . . , B˜m := B(cm , εm )\{cm } do not contain any a-points of f . Since f has no a-points and no

34

Chapter 1. Function theory in hypercomplex spaces

singularities in E\

m

B˜i , one obtains consequently  1 q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)]

i=1

Ak+1

Γ

=

1

i=1

Ak+1



=



m 

q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)]

∂B(ci ,εi )

ord(f − a; c).

c∈E




Remark. Recall that the formula (1.38) simplifies to dσ(f (z)) = det Jf (z)[((Jf )−1 )tr (z)] ∗ [dσ(z)]

(1.41)

whenever f is a local diffeomorphism, where tr means to take the transpose of the matrix. See [165] for instance. As a consequence of this, we can then rewrite the argument principle (1.40) in the form   1 ord(f − a; c) = q0 (f (z) − a) det Jf (z)[((Jf )−1 )tr (z)] ∗ [dσ(z)] (1.42) Ak+1 c∈E

Γ

whenever Γ is a cycle that has at most a countable number of points at which det Jf (z) = 0. The argument principle allows us to set up a generalized version of Rouch´e’s theorem for monogenic functions with isolated zeroes. In [74] we proved: Theorem 1.35 (Generalized Rouch´e’s Theorem for isolated zeroes). Let G ⊂ Ak+1 be a domain and let Γ be a nullhomologous k-dimensional cycle parameterizing a k-dimensional surface of a (k + 1)-dimensional simply connected domain E ⊂ G. Let us assume that f, g : G → Ak+1 are monogenic functions that have both only a finite number of zeroes in E and no zeroes on the boundary ∂E. If

f (z) − g(z) < f (z) then

 c∈E

ord(f ; c) =



for all z ∈ ∂E,

(1.43)

ord(g; c).

(1.44)

c∈E

Proof. Let λ ∈ [0, 1]. Next define the function hλ := f + λ(g − f ) and consider  ord(hλ ; c) N (λ) := c∈E

=

1



Ak+1 Γ

q0 (f (z) + λg(z) − λf (z))[(J(f + λg − λf ))∗ ] ∗ [dσ(z)].

1.6. The generalized negative power functions

35

The integrand depends continuously on λ. N (λ) is thus a continuous function. However, N (λ) ∈ Z, so that N (λ) does not depend on λ. Therefore, N (λ) is a constant, and consequently, N (1) = N (0). The proof is hereby finished.
With special monogenic automorphic forms which will be described in the following chapters, one can extend these techniques to the more general framework of conformally flat spin manifolds. In Chapter 2.11 we will present argument principles and Rouch´e type theorems explictly on conformally flat cylinders and tori.

1.6

The generalized negative power functions (s)

The generalized negative power functions qn (z) provide elementary building blocks for the construction of higher dimensional Clifford-valued monogenic, Euclidean harmonic and polymonogenic Eisenstein and Poincar´e series and related variants of Riemann zeta type functions. Adequate representation formulas for these functions are needed for a quantitative understanding and description of the associated function classes that will be developed in the following two chapters. In [13] and [37] one can find some representation formulas of the monogenic functions qn (z) in terms of an infinite series of Lagrange or Gegenbauer polynomials, respectively. However, they do not give a description of the functions qn in terms of a finite explicit sum of explicitly determined functions. In the recent papers [86] and [23] new and essentially different formulas were developed which meet most of our ends for the monogenic case. In [86] we developed an explicit recurrence formula in terms of finite permutational products of 2k hypercomplex variables. This formula leads nearly immediately to an impor(1) tant fully explicit upper bound estimate on the monogenic qn functions. This estimate in turn will provide the central tool for the convergence analysis of the related Eisenstein and Poincar´e series. In [23] also a non-recurrent finite closed representation formula for the func(1) tions qn is developed in particular. By means of this formula, we will later be able to give a fully explicit description of the variants of Riemann zeta functions which appear in the framework of monogenic and polymonogenic automorphic forms and functions. In [86, 23] these formulas have been established in the context of 1-monogenic functions in real Euclidean spaces. In [89] we have seen that they extend naturally to the framework of complexified Clifford analysis, although there asymptotic and growth behavior is different from the real Euclidean case. This is due to the fact that the functions qn have in the complexified case a singularity cone in the space, and not only one isolated singularity at the origin. This behavior is then reflected in the corresponding estimate on the qn -functions, which is more complicated in the complexified case than in the case of real Euclidean space. One obtains a similar result for Minkowski type

36

Chapter 1. Function theory in hypercomplex spaces

spaces with arbitrary signature (p, q) where both p and q are different from zero. This case can be treated in complete analogy to the complexified case discussed in [89]. In this section we will stick to the setting in real Euclidean space. In addition to [86] and [23] we will provide here a detailed description of the s-monogenic negative power functions for s < k + 1, too. Notice that in the case s > 1, the (s) functions qn with n = (n0 , . . . , nk ) where n0 > 0, are in general not linear (s) combinations of qn functions with n0 = 0. Therefore, in addition to [86, 23], (s) we also develop representation formulas for qn with n0 > 0. We start with the treatment of the cases where s is odd, which includes in particular the monogenic case s = 1. We begin by showing Lemma 1.36. Assume that s is an odd positive integer with s < k + 1. Let n ∈ IN. Then n−1 

∂ n (s) q (z) = ∂xn1 0

j=0

n−1 (s) j! q(n−(j+1))τ (1) (z) j ' ( k − s −1 j+1 k+2−s )(z e1 ) )(e1 z −1 )j+1 . · ( + (−1)j+1 ( 2 2

Proof. We prove this lemma by induction. A direct computation gives (s)

qτ (1) (z)

e1 ze1 z e1 = − z k+2−s + k+2−s − k+2−s 2 2 z k+2−s z k+4−s  (s) −1 = q0 (z) k−s e1 ) − k+2−s (e1 z −1 ) . 2 (z 2

The assertion holds for n = 1. Now let n ≥ 1. For n ∈ IN we obtain by induction ∂n {z −1 } ∂xn 1

= n!(z −1 e1 )n z −1

and

∂n {z −1 } ∂xn 1

= (−1)n n!z −1 (e1 z −1 )n .

(1.45)

Hence, (s)

q(n+1)τ (1) (z) =

∂ ∂x1

n−1  j=0

n−1 j



(s)

j! q(n−(j+1))τ (1) (z)

 −1 e1 )j+1 + (−1)j+1 ( k+2−s )(e1 z −1 )j+1 · ( k−s 2 )(z 2 n−1   n−1  (s) −1 j! q(n−j)τ (1) (z) ( k−s = e1 )j+1 + (−1)j+1 ( k+2−s )(e1 z −1 )j+1 j 2 )(z 2 j=0  (s) −1 e1 )j+2 +q(n−(j+1))τ (1) (z) ( k−s 2 )(j + 1)(z  +(−1)j+2 ( k+2−s )(j + 1)(e1 z −1 )j+2 2

1.6. The generalized negative power functions =

n−1  j=0 n 

n−1 j

+

j=1

=

j=0

 (s) −1 j! q(n−j)τ (1) (z) ( k−s e1 )j+1 + (−1)j+1 ( k+2−s )(e1 z −1 )j+1 2 )(z 2

n−1 j−1

n    n j



37

 (s) −1 j! q(n−j)τ (1) (z) ( k−s e1 )j+1 + (−1)j+1 ( k+2−s )(e1 z −1 )j+1 2 )(z 2

 (s) −1 j! q(n−j)τ (1) (z) · ( k−s e1 )j+1 + (−1)j+1 ( k+2−s )(e1 z −1 )j+1 . 2 )(z 2


The lemma is hereby proven. n

(s)

∂ For the functions ∂x with i = 2, . . . , k we obtain an analogous recurrence n q0 i formula. We only have to replace the element e1 by ei . In view of

∂ n −1 ∂ n −1 {z } = n!(z −1 (−e0 ))n z −1 and {z } = (−1)n n!z −1 (e0 z −1 )n , (1.46) n ∂x0 ∂xn1 n

(s)

∂ we get a slightly different formula for ∂x n q0 (z). This is an effect of the symmetry 0 break in the paravector formalism in which the x0 -direction is distinguished. If we perform the same calculations as in the previous lemma, then we arrive at

Lemma 1.37. Assume that s is an odd positive integer with s < k + 1. Let n ∈ IN. Then n−1  n − 1 ∂ n (s) (s) q (z) = j! q(n−(j+1))τ (0) (z) j ∂xn0 0 j=0 ' ( k − s −1 k+2−s )(z (−e0 ))j+1 + (−1)j+1 ( )(e0 z −1 )j+1 . · ( 2 2

The next step is to develop such a recurrence formula, where n is a general \{0}. multi-index of INk+1 0 First we treat all multi-indices (n0 , n1 , . . . , nk ) where n0 = 0. If we deal with mixed derivatives, then permutational products appear. We will illustrate this by the following examples. We assume that i, j ∈ {1, . . . , k} are pairwise distinct. Then we obtain ' ( ∂ ∂ (s) k − s −1 k+2−s (s) −1 )(z ei ) − ( )(ei z ) q (z) = qτ (j) (z) ( ∂xj ∂xi 0 2 2 ' ( k − s −1 k+2−s (s) −1 −1 −1 + q0 (z) ( )(z ej )(z ei ) + ( )(ei z )(ej z ) . 2 2

38

Chapter 1. Function theory in hypercomplex spaces

Furthermore, (s)

∂ 3 q0 (z) ∂xj ∂x2i

' ( k − s −1 k+2−s (s) )(z ei ) − ( )(ei z −1 ) = qτ (j)+τ (i) (z) ( 2 2 ' ( k − s k +2−s (s) + qτ (i) (z) ( )(z −1 ej )(z −1 ei ) + ( )(ei z −1 )(ej z −1 ) 2 2 ' ( k − s k + 2 − s (s) + qτ (j) (z) ( )(z −1 ei )2 + ( )(ei z −1 )2 2 2  k−s (s) + 2 q0 (z) ( )(z −1 ej ) × (z −1 ei ) · (z −1 ei ) 2  k+2−s )(ei z −1 ) · (ei z −1 ) × (ej z −1 ) . −( 2

For the iteration of this procedure we deduce by a simple induction proof the following formulas: ∂ {[z −1 ei ]ni × [z −1 ej ]nj · (z −1 ej )}, ∂xj = (ni + nj + 1)[z −1 ei ]ni × [z −1 ej ]nj +1 · (z −1 ej ) ∂ {(ei z −1 ) · [ei z −1 ]ni × [z −1 ej ]nj } ∂xj = −(ni + nj + 1)(ei z −1 ) · [ei z −1 ]ni × [ej z −1 ]nj +1 .

(1.47)

With these formulas in hand we can finally show the following theorem using the nk n1  n2    notation := ··· : 0≤j≤n

j1 =0 j2 =0

jk =0

Theorem 1.38. Suppose that s is an odd integer with s < k + 1. Let n := (n0 , n1 , n2 , . . . , nk ) ∈ INk+1 with n0 = 0. Then 0 (s)

qn+τ (1) (z) =

 n (s) |j|! qn−j (z) j 0≤j≤n  k−s )[z −1 ek ]jk × · · · × [z −1 e1 ]j1 · (z −1 e1 ) · ( 2  k+2−s )(e1 z −1 ) · [e1 z −1 ]j1 × · · · × [ek z −1 ]jk . +(−1)|j|+1 ( 2

Proof. For multi-indices of the form n := (0, n1 , 0, . . . , 0) with n1 ∈ IN0 , the statement has been proved in Lemma 1.36. Let us now assume that n is a multi-index of the form n := (0, n1 , n2 , 0, . . . , 0) where n2 ∈ IN0 \{0}. By a direct computation one verifies immediately that the assertion is true for n2 = 1. In the sequel we assume that n2 ≥ 1 and compute

1.6. The generalized negative power functions

39

(s) ∂ ∂x2 q0,n1 +1,n2 ,0,...,0 (z)

=

 n j

0≤j≤n

(s)

|j|!q0,n1 −j1 ,n2 +1−j2 ,0,...,0 (z)

 −1 e2 ]j2 × [z −1 e1 ]j1 · (z −1 e1 ) · ( k−s 2 )[z

+

+ (−1)|j|+1 ( k+2−s )(e1 z −1 ) · [e1 z −1 ]j1 × [e2 z −1 ]j2 2  n (s) j |j|!q0,n1 −j1 ,n2 −j2 ,0,...,0 (z)



0≤j≤n

 −1 · (|j| + 1) ( k−s e2 ]j2 +1 × [z −1 e1 ]j1 · (z −1 e1 ) 2 )[z

=

+(−1)|j|+2 ( k+2−s )(e1 z −1 ) · [e1 z −1 ]j1 × [e2 z −1 ]j2 +1 2  n (s) j |j|!q0,n1 −j1 ,(n2 +1)−j2 ,0,...,0 (z)



0≤j≤n

 −1 · ( k−s e2 ]j2 × [z −1 e1 ]j1 · (z −1 e1 ) 2 )[z +(−1)|j|+1 ( k+2−s )(e1 z −1 ) · [e1 z −1 ]j1 × [e2 z −1 ]j2 2

j2 ←j2 +1

+

 τ (2)≤j≤n+τ (2)



n j−τ (2)



(s)

|j|!q0,n1 −j1 ,n2 +1−j2 ,0...,0 (z)

 −1 · ( k−s e2 ]j2 × [z −1 e1 ]j1 · (z −1 e1 ) 2 )[z

=



+(−1)|j|+1 ( k+2−s )(e1 z −1 ) · [e1 z −1 ]j1 × [e2 z −1 ]j2 2 n+τ (2)  (s) |j|!q0,n1 −j1 ,(n2 +1)−j2 ,0,...,0 (z) j



0≤j≤n+τ (2)

 −1 · ( k−s e2 ]j2 × [z −1 e1 ]j1 · (z −1 e1 ) 2 )[z

 −1 −1 j1 −1 j2 . +(−1)|j|+1 ( k+2−s )(e z ) · [e z ] × [e z ] 1 1 2 2

The assertion turns out to be true for (n2 + 1). Hence, the formula is proved for all indices n of the form (0, n1 , n2 , 0, . . . , 0) with (n1 , n2 ) ∈ IN20 . Next we assume that n has the form (0, n1 , n2 , n3 , 0, . . . , 0), with (n1 , n2 , n3 ) ∈ IN30 . For multi-indices n := (0, n1 , n2 , n3 , 0, . . . , 0) with n3 = 0 the validity of the formula has been proved by the previous induction step. With induction over n3 one verifies, analogously to the previous induction procedure, that the formula holds for all multi-indices of the form (0, n1 , n2 , n3 , 0, · · · , 0). Subsequently, one proceeds to consider multi-indices n of the form (0, n1 , n2 , n3 , n4 , 0, . . . , 0), establishes the formula by induction over n4 analogously as we have shown in the second step, and proceeds with further induction steps until one finally obtains
the validity of the formula for all multi-indices (0, n1 , n2 , . . . , nk ) ∈ INk0 .

40

Chapter 1. Function theory in hypercomplex spaces

\{0} is an index with n1 = 0, one In the case where (0, n1 , n2 , . . . , nk ) ∈ INk+1 0 chooses an α ∈ {2, . . . , k} with mα = 0. In the associated formula for the function (s) qn the index α plays then the role of the index 1 in Theorem 1.38, so that we get more generally: Corollary 1.39. Let s be an odd integer with s < k + 1. Let α ∈ {1, . . . , k} and n ∈ INk+1 with n0 = 0. Then 0  n (s) (s) qn+τ (α) (z) = |j|! qn−j (z) j 0≤j≤n  k−s )[z −1 ek ]jk × · · · × [z −1 e1 ]j1 · (z −1 eα ) · ( 2  k+2−s )(eα z −1 ) · [e1 z −1 ]j1 × · · · × [ek z −1 ]jk . +(−1)|j|+1 ( 2 Now we see how to extend this formula to the general case involving also indices with n0 = 0. If n = (n0 , n1 , . . . , nk ) is a general index from INk+1 \{0} 0 with n1 = · · · = nk = 0, then we are in the situation of Lemma 1.37. Let us thus suppose that there is at least one α ∈ {1, . . . , k} for which nα = 0. We can hence write  ∂ n0  (s) q (z) qn(s) (z) = ∂xn0 0 m+τ (α) where m is a multi-index with m0 = 0. For qm+τ (α) (z) Corollary 1.39 provides us with a recurrence formula. Thus we need only to perform differentiations with respect to x0 on the formula in Corollary 1.39. Applying (1.46) to Corollary 1.39, then an analogous induction argument leads to the final result: Theorem 1.40. Let s be an odd positive integer with s < k + 1. Let α ∈ {1, . . . , k} and n = (n0 , . . . , nk ) ∈ INk+1 . Then 0 (s)

qn+τ (α) (z)

 n (s) = |j|! qn−j (z) j 0≤j≤n  k−s )[z −1 ek ]jk × · · · × [z −1 e1 ]j1 × [z −1 (−e0 )]j0 · (z −1 eα ) · ( 2  k+2−s )(eα z −1 ) · [e0 z −1 ]j0 × [e1 z −1 ]j1 × · · · × [ek z −1 ]jk . +(−1)|j|+1 ( 2

All the configurations of multi-indices n are now treated for the cases where s is odd. Notice that the terms with k−s 2 in Theorem 1.40 vanish if k = s and one obtains a much simpler formula. In the case where s is odd and where additionally (k) s = k, we are in the holomorphic Cliffordian case whence the functions qn (z) simplify to the partial derivatives of z −1 . In view of (1.47) we thus obtain in this

1.6. The generalized negative power functions

41

special case the holomorphic Cliffordian negative basis polynomials from [104] which read in our notation: [z −1 e0 ]n0 × [z −1 e1 ]n1 × · · · × [z −1 ek ]nk z −1 . Let us now turn to the cases where s is even. We shall see that we can treat these (s) cases rather similarly. Notice that for even s the functions qn are all scalar-valued. To derive similar formulas for even s with s < k +1, let us again first consider is a multi-index of the form nτ (i) where i ∈ {1, . . . , k}. the case where n ∈ INk+1 0 We compute 1 ∂ (s) qτ (i) (z) = 2 ∂xi x0 + · · · + x2k (k−s+1)/2 1 (k − s + 1)xi = − 2 2

z (x0 + · · · + x2k )(k−s+1)/2

 k − s + 1 (s) q0 (z) z −1 ei − ei z −1 , = (1.48) 2 in view of xi = − 12 (zei + ei z) for i ∈ {1, . . . , k}. If we next apply on (1.48) the formulas (1.45), then we arrive by an absolutely analogous induction argument as in Lemma 1.36 at the following representation formula: Lemma 1.41. Let s ∈ 2IN with s < k + 1. Let i ∈ {1, . . . , k}. Then for all n ∈ IN, n−1  n − 1 (s) qnτ (i) (z) = j!q(n−(j+1))τ (i) (z) j j=0 

k − s + 1  (z −1 ei )j+1 + (−1)j+1 (ei z −1 )j+1 . · 2 Turning to the case i = 0, then we observe first that (s)

qτ (0) (z) = −

(k − s + 1)x0

z 2

1

(k−s+1)/2 x20 + · · · + x2k

z z  k − s + 1 (s) q0 (z) = (−1) · + 2 2

z

z 2   k − s + 1 (s) −1 q0 (z (−e0 )) + (−1)(e0 z −1 ) , (1.49) = 2 in view of x0 = 12 (z + z). Applying next on (1.49) the formulas (1.46), then an analogous induction argument as in Lemma 1.37 leads finally to Lemma 1.42. Let s ∈ 2IN with s < k + 1. Then for all n ∈ IN, n−1  n − 1 (s) qτ (0) (z) = j!q(n−(j+1))τ (0) (z) j j=0 

k − s + 1  (z −1 (−e0 ))j+1 + (−1)j+1 (e0 z −1 )j+1 . · 2

42

Chapter 1. Function theory in hypercomplex spaces

In complete analogy to the proof of Theorem 1.40 one can deduce with an absolutely analogous induction argument, by applying the differentiation formulas (1.47) on the expressions in Lemma 1.41 and in Lemma 1.42, the following general (s) recurrence formula for the functions qn for even s with s < k + 1: Theorem 1.43. Let s be an even positive integer with s < k + 1. Let α ∈ {1, . . . , k} . Then and n = (n0 , . . . , nk ) ∈ INk+1 0   n (s) (s) qn+τ (α) (z) = |j|!qn−j (z) j 0≤j≤n

k − s + 1  [z −1 ek ]jk × · · · × [z −1 e1 ]j1 × [z −1 (−e0 )]j0 · (z −1 eα ) · 2  +(−1)|j|+1 (eα z −1 )[e0 z −1 ]j0 × [e1 z −1 ]j1 × · · · × [ek z −1 ]jk .

We proceed to show that we can now easily derive from Theorem 1.40 and (s) Theorem 1.43 the following estimates on the functions qn . the following Proposition 1.44. Let s ∈ {1, . . . , k}. For all multi-indices n ∈ INk+1 0 estimate holds for all z ∈ Ak+1 \{0}:  (k + 1 − s)(k + 2 − s) · · · (k + |n| − s)  ∂ |n|   (s) . (1.50)  n q0 (z) ≤ ∂z

z k+|n|+1−s Proof. We first restrict ourselves to the cases where n = (0, n1 , 0, . . . , 0). A simple calculation shows that the estimate holds for n1 = 0 and n1 = 1, both for even and odd s. In the sequel we assume n1 ≥ 1. We obtain both for even and odd s:  n +1    n1 ∂ 1  (s) ((k−s)+(n1 −j))!  n1 +1 q0 (z) ≤ z k+1−s k+n1 +2−s n1 ! (k−s)!(n1 −j)! ∂x1

j=0

=

k+1−s z k+n1 +2−s

n1 !

 n1   (k−s)+n1 −j j=0

n1 −j

k+1−s+n1 

=

k+1−s z k+n1 +2−s

=

(k+n1 +1−s)! k+1−s z k+n1 +2−s (k+1−s)!

=

(k + 1 − s)(k + 2 − s) · · · (k + n1 + 1 − s) z k+n11 +2−s .

n1 !

n1

Thus, the assertion holds for indices of the form n := (0, n1 , 0, . . . , 0). With the estimates     (e1 z −1 ) · [e0 z −1 ]j0 × [e1 z −1 ]j1 × · · · × [ek z −1 ]jk  ≤ e1 z −1 |j|+1 ,   −1 j   [z ek ] k × · · · × [z −1 e1 ]j1 × [z −1 (−e0 )]j0 · (z −1 e1 ) ≤ z −1 e1 |j|+1 and the formula

 n0 nk |n| ··· = j j0 jk k+1

j∈IN0 |j|=j

1.6. The generalized negative power functions

43

in combination with the statements of Theorem 1.40 and Theorem 1.43, respectively, we get furthermore, by applying a simple induction argument, that (s)

qn(s) (z) ≤ q0,|n|,0,...,0 (z)

∀n ∈ INk+1 0


and the assertion is shown.

Remarks. This formula provides actually a more precise estimate than that given in [13]. The estimate in (1.50) is furthermore stronger than the estimate proved in [63] by R. Fueter for the quaternionic case in the framework s = 1, i.e.,

qn(1) (z) ≤ (|n| + 2)! z −(|n|+3) .

(1.51)

R. Fueter’s method for his proof is based on the formula qn(1) (ζ) = ζ −1

∂ |n| [∆z {(zζ −1 )n+2 }], ∂xn

z ∈ IH, ζ ∈ IH\{0},

(1.52)

where ∆z denotes the Laplace operator with respect to the variable z. Switching from the complex holomorphic case, to the higher dimensional 1-monogenic cases, then the ordinary positive power functions are replaced by permutational products of the hypercomplex variables Zi = xi − ei x0 . See [58], and in particular [118, 120]. Theorem 1.40 provides us with a similar representation formula of the negative power functions that is built with permutational products of the special hypercomplex variables ζi := z −1 ei and ηi := ei z −1 . Theorem 1.40 and Theo(s) rem 1.43 are explicit recurrence formulae for the functions qn (z). (s)

In the vector formalism where the analogues of the functions qn are given by q(s) n1 ,...,nk (x) =

∂ n1 +···+nk x1 e1 + · · · + xk ek (s) (s) , nk q0 (x) where q0 (x) = − 2 n1 ∂x1 · · · ∂xk |x1 + · · · + x2k |(k+1−s)/2

one obtains the following analogue of Theorem 1.40: Theorem 1.45. Let s be an odd integer with s < k. Let α ∈ {1, . . . , k} and n ∈ INk0 \{0}. Then for all x ∈ IRk \{0} we have  n (s) (s) qn+τ (α) (x) = (1.53) |j|! qn−j (x) j 0≤j≤n  k−1−s )[x−1 ek ]jk × · · · × [x−1 e1 ]j1 · (x−1 eα ) · (−1)|j|+1 ( 2  k+1−s (eα x−1 ) · [e1 x−1 ]j1 × · · · × [ek x−1 ]jk . + 2

44

Chapter 1. Function theory in hypercomplex spaces

The proof can be done in analogy to the proof of Theorem 1.40. The additional symmetry simplifies several computations. Similarly, one obtains the follow(s) ing formula for the functions qn for the cases where s is an even positive integer with s < k: Theorem 1.46. Let s be an odd integer with s < k. Let α ∈ {1, . . . , k} and n ∈ INk0 \{0}. Then for all x ∈ IRk \{0} we have  n (s) (s) (1.54) qn+τ (α) (x) = |j|! qn−j (x) j 0≤j≤n

k − s  [x−1 ek ]jk × · · · × [x−1 e1 ]j1 · (x−1 eα ) · (−1)|j|+1 2  + (eα x−1 ) · [e1 x−1 ]j1 × · · · × [ek x−1 ]jk .

Remark. These representation formulas hold also in its form for the analogues of (s) qm in the more general framework of arbitrary real and complex Minkowski type spaces of signature (p, q). However, one gets a different form of estimate due to the infinite extension of the singularity cone. For details, see [89] where we developed an estimate for the complexified case. (s)

Next we describe a closed representation formula for the functions qn in Ak+1 that was developed in [23] for the monogenic case s = 1. The formula that (s) will be derived is based on the fact that the functions qn (z) are radial symmetric functions of degree −(k − s + 1) whenever s is even, and that further to this the special relation 1 Dz qn(s) (z) (1.55) qn(s−1) (z) = − k−s+1 (s)

holds. The whole problem of finding a closed formula for the functions qn is thus reduced to finding a closed description for partial derivatives of functions in the space that have a radial symmetry. To meet this end one needs the following lemma: Lemma 1.47. Let f (r) be a C ∞ function of a single real variable r. Then 

d dr

n



f (r) =

0≤2p≤n

(2r)n−2p n! (n − 2p)!p!



d d(r2 )

n−p f (r).

(1.56)

Proof. By applying the chain rule from classical one real variable calculus, one obtains directly that  d  1 d d(r 2 ) f (r) = 2r dr f (r), which gives



d dr



f (r) = 2r

d d(r 2 )

 f (r).

1.6. The generalized negative power functions

45

Iteration leads in the second step to 

 d 2 dr

f (r) = 2

d d(r 2 )



f (r) + 4r2



d d(r 2 )

2

f (r).

Arbitrary iterations lead finally to a formula of the form 

 d n dr

f (r) =

∞ 

cn,s (r)

s=0

d d(r 2 )

s f (r),

(1.57)

in which however only a finite number of terms do not vanish. It is thus only a finite sum. The main task is reduced to determining the functions cn,q (r) explictly. 2 To do so we apply the following trick: insert the particular function f (r) = ear into (1.57). A direct computation leads to ∞  d n ar2  2 s e = c (r)a (1.58) ear . n,s dr s=0

The functions cn,s (r) will now be determined by a comparison of both sides of the previous equation. Next we multiply both sides of equation (1.58) by bn /n! and sum over n = 0, 1, 2, . . ., which then leads to ∞  n=0

bn n!



d dr

n

∞  ∞ 

2

ear =

n=0

s=0

 2 n cn,s (r)as ear bn! .

The left-hand side of this equation is nothing else than the Taylor expansion of 2 the function h(b) = ea(r+b) at the point b = 0, so that this equation simplifies to ) * ∞  2 as bn a(r+b)2 = cn,s (r) n! (1.59) ear , e n,s=0

which in turn simplifies further to eab(2r+b) =

∞ 

s n

n,s=0

cn,s (r) a n!b .

(1.60)

Now we rewrite eab(2r+b) in terms of its Taylor expansion around r = 0, and we get ∞ q q ∞   s n a b (2r+b)q = cn,s (r) a n!b . (1.61) q! q=0

n,s=0

In view of the binomial formula (b + 2r)q = the previous equation in the form  0≤p≤q k,    Sc qn(s) (z)  ≡ 0 z=x0

if and only if n = (n0 , . . . , nk ) is a multi-index from INk+1 with n1 , . . . , nk ∈ 2IN0 . 0

Chapter 2

Clifford-analytic Eisenstein series associated to translation groups 2.1

Multiperiodic Mittag-Leffler series

The classical meromorphic Eisenstein series for translation groups in complex analysis are given as Mittag-Leffler series of the negative power functions 1/(z + ω)m summed over a one- or two-dimensional lattice. They provide important building blocks for the construction of the trigonometric functions, the elliptic functions and the elliptic modular forms. In the higher dimensional Clifford analysis setting where we consider nullsolutions to D[s] or Ds in Ak+1 or IRk , respectively, the negative power functions (s) are generalized by the functions qm (z + ω), as explained in the previous chapter. Generalizations of the classical Eisenstein series to the context considered here (s) are, roughly speaking, thus given by summations of the expressions qm (T z ) over the corresponding translation group T . The following definition provides a more precise formulation: Definition 2.1 (s-monogenic Eisenstein series associated to translation groups). Let p ∈ IN with 1 ≤ p ≤ k + 1 and let s ∈ IN with 1 ≤ s ≤ k. Assume that ω1 , . . . , ωp are IR-linear independent paravectors from Ak+1 . Then the s-monogenic translative Eisenstein series associated with the p-dimensional lattice with |m| ≥ Ωp = Zω1 + · · · + Zωp are defined for all multi-indices m ∈ INk+1 0 max{0, p − k + s} by  (s)
(p) qm (z + ω). (2.1) m,s (z; Ωp ) = ω∈Ωp

50

Chapter 2. Clifford-analytic Eisenstein series for translation groups

In the cases where p−k−1+s ≥ 0 we have additionally for multi-indices m ∈ INk+1 0 with |m| = p − k − 1 + s:  

(s) (s) (s) q (z; Ω ) = q (z) + (z + ω) − q (ω) . (2.2)
(p) p m,s m m m ω∈Ωp \{0}

In the cases where p − k − 2 + s ≥ 0 one defines, for multi-indices m ∈ INk+1 with 0 |m| = p − k − 2 + s, (p)


m,s;a,b (z; Ωp ) := +

(s) (s) qm (z − a) − qm (z − b)   (s) (s) (z − a + ω) − qm (z − b + ω) qm ω∈Ωp \{0}

(2.3)

 (s) (s) − qm (ω − a) + qm (ω − b) ,

where a, b are paravectors from Ak+1 \Ωp with a ≡ b mod Ωp . Notice that all these series have only point singularities. This stems from the fact that we are working here in real Euclidean spaces which are endowed with a definite scalar product. As shown in [89, 90, 93] one can also introduce these function series in the more general context of real and complex Minkowski type spaces IRp,q and Cp,q endowed with a non-degenerate scalar product of arbitrary signature (p, q) with p + q = k + 1. However, in these cases one has to put restrictions on the generators of the lattice. In those quadratic spaces IRp,q where p and q are both different from zero, one does not always obtain a convergent series when considering a summa(s) tion of the associated expressions qm (z + ω) over any arbitrary lattice in IRp,q , independently how large |m| is chosen. This behavior stems from the fact that these expressions do not only have singularities in isolated points but in cones. Therefore, restrictions on the periodicity group have to be made in these cases. One obtains a convergent series (under the same conditions for |m| as mentioned above), when the lattice Ω is additionally completely contained in IRp or in IRq . This shows that the configuration (p, q) = (0, k +1) or (p, q) = (k +1, 0) is a special one. In these particular cases one obtains an even s-monogenic Eisenstein series associated to periodicity lattices of the codimension 0. In the general case working in IRp,q , one gets in the best case an Eisenstein series associated to a translation group with max{p, q} linear independent generators. In the complexified case working in Cp,q one has again a more balanced relationship between the corresponding spaces. In Cp,q one can always define convergent Eisenstein series to a group that has k + 1 translation generators. In Cp,q the period lattice needs to be contained in the linear IRk+1 -like domain manifold eiφ (IRp + iIRq ) where φ ∈ [0, 2π) is an arbitrary real parameter. Under this condition we get the same convergence conditions for |m| as mentioned above. Thus, in

2.1. Multiperiodic Mittag-Leffler series

51

the complexified case one can always introduce (independently from the signature) a p-fold periodic s-monogenic Eisenstein series, for 1 ≤ p ≤ k + 1. One obtains a divergent series when p > k + 1. In the cases p < k +1 these function series provide us with building blocks for k-fold periodic generalizations of classical trigonometric functions to the Clifford analysis setting (cf. [84, 85]). In the case p = k + 1, one obtains generalizations of the elliptic functions to the Clifford analysis setting. Notice that one cannot define (k + 1)-fold periodic s-monogenic Eisenstein series within the setting of IRp,q whenever p, q are both different from zero. (k+1)

The particular 1-monogenic function series
m,1 , where the summation is extended over a period lattice of codimension 0, appears first in a paper by A. C. Dixon [40] within the particular framework of the three dimensional real Euclidean space IR0,3 , some decades later in works of R. Fueter [61, 62, 63, 64] in the setting of real quaternions, and in the beginning of the 1980s in J. Ryan’s first paper [133] in the setting of the (k + 1)-dimensional real Euclidean vector space. Some of their basic properties were studied in these works. In [128] T. Qian considered a class of functions defined on particular sector domains and studied Fourier multipliers and Lipschitz perturbations of the k(k) torus in IRk+1 within a cylinder. The particular function
0 associated with the k+1 turns out to be part of that special k-dimensional orthonormal lattice in IR function class. In [84, 85] we started a more extensive and systematic study of 1-monogenic Eisenstein series associated to arbitrarily dimensional translation groups in the Euclidean space IR0,k+1 within a more general framework, particularly under function theoretical and number theoretical motivation. Subsequently their complexification to C0,k+1 has been studied in [89]. Polymonogenic Eisenstein and Poincar´e series associated to general discrete subgroups of Vahlen type groups in IR0,k+1 have been introduced in [88] and their complexifications to C0,k+1 in [89]. The function series treated in [88] contain the series (p) (p)
0,s as special subseries. Explicit analogues of the series
0,s on the unit ball in IR0,k+1 and on the Lie ball in C0,k+1 also were developed in [88] and [90], respec(p) tively. In [96] the series
0,s with p ≤ min{k + 2 − s, k + 1} were used explicitly to develop Cauchy–Green formulas on conformally flat cylinders and tori which arise from factoring out IR0,k+1 by a translation group. For arbitrary multi-indices (p) m ∈ INk+1 and s < k + 1 the series
m,s were introduced in [92] within the 0 framework of real Euclidean spaces. The family of all s-monogenic Eisenstein series associated to translation groups in turn contains the holomorphic Cliffordian Weierstraß functions from [67, 105], which are solutions of the Fueter–Sce equation, as very special and important subcases. Finally, in [93] we proceeded also to introduce s-monogenic generalizations of Eisenstein series within the framework of finite dimensional real and complex vector spaces.

52

Chapter 2. Clifford-analytic Eisenstein series for translation groups

We restrict ourselves mainly to a treatment of the theory within the context of finite dimensional real Euclidean vector spaces. For extensions of this theory to complexified Clifford analysis, we refer to our papers [89] and [90], and, more generally, for extensions to the framework of real and Minkowski type spaces of arbitrary signature to [93]. (p)

The Eisenstein series
m,s (z, Ωp ) provide an extremely useful tool for the construction of large classes of s-monogenic p-fold periodic functions in Ak+1 . As we shall see in Chapter 3, they turn out to serve furthermore as building blocks for automorphic forms for much larger arithmetic groups which contain translation groups as subgroups. This gives them a similar key role as their two-dimensional counterparts from classical complex analysis have. We begin by giving a detailed convergence proof for these function series under the conditions mentioned in Definition 2.1. Proposition 2.2. Under the conditions for p, s, m stated in Definition 2.1, the s-monogenic generalized translative Eisenstein series converge normally in Ak+1 \Ωp . Proof. Let us first consider those cases where m is a multi-index from INk+1 with 0 |m| ≥ max{0, p − k + s}. To show that under these conditions the series 

(s)

qm (z + ω)

(2.4)

ω∈Ωp

is normally convergent in Ak+1 let us take an arbitrary compact subset K ⊂ Ak+1 . Next one takes a real R > 0 so that the compact ball B(0, R) covers K completely. Let z ∈ B(0, R). For the qualitative convergence analysis we may drop without loss of generality a finite number of terms of the series. Let us thus restrict ourselves to summing those lattice points ω that satisfy

ω > (k + 1)R ≥ (k + 1) z .

(2.5)

(s) g(z) = qm (z + ω)

(2.6)

The function is left s-monogenic in 0 ≤ z < (k+1)R. Hence it is real analytic in B(0, (k+1)R) and is thus represented in the interior of this ball by its Taylor series which turns out to have the form (s)

qm (z + ω) =

∞  |l|=0

1 l0 l! x0

(s)

· · · · · xlkk qm+l (ω).

(2.7)

As a consequence of the estimate proved in Proposition 1.44 we obtain in particular    ∞   (s) qm (z + ω) ≤



l=0 l=l0 +···+lk

l  |m|+k−s  + Rl  l! 1 (l0 + · · · + lk + γ) l0 !···l .  ω k+|m|+1−s l k ! w γ=1

2.1. Multiperiodic Mittag-Leffler series

53

An application of the multinomial formula  l! l=l0 +···+lk

leads finally to     (s) qm (z + ω) ≤

l0 !···lk !

= (k + 1)l

l  ∞  |m|+k−s  +   1 (l + γ)  (k+1)R  ω k+|m|+1−s ω γ=1

l=0

=

(|m|+k−s)! 1 (k+1)R k+|m|+1−s ω k+|m|+1−s 1− ω

≤

C ω k+|m|+1−s

with a real positive constant C. Due to G. Eisenstein’s lemma (cf. [41]) a series of the form 

m1 ω1 + · · · + mt ωt −(t+α) (2.8) (m1 ,...,mt )∈Zt \{0}

is convergent if and only if α ≥ 1. Hence, the series (2.4) converges normally for |m| ≥ max{0, p − k + s}. is a multi-index Let us now turn to the second type of cases, where m ∈ INk+1 0 with |m| = p − k − 1 + s while we suppose that p − k − 1 + s ≥ 0. In this context, where we have to show that the expression   (s) (p) (s) (s) qm (z + ω) − qm (ω) (2.9)
m,s (z; Ωp ) = qm (z) + ω∈Ωp \{0}

is normally convergent in Ak+1 \Ωp , we can provide a similar argument. Again we suppose that z ∈ B(0, R) and restrict our study to those lattice points with

ω > (k + 1)R ≥ (k + 1) z . Instead of considering the function (2.6), we expand the function (s) (s) (z + ω) − qm (ω), g1 (z) = qm

|m| = p − k − 1 + s,

into a Taylor series in B(0, R ) where 0 < R < R. We obtain g1 (z) =

∞  |l|=1

1 l0 l! x0

(s)

· · · · · xlkk qm+l (ω).

(2.10)

Notice that in contrast to (2.7), this Taylor series starts only with |l| ≥ 1. A similar procedure, using the estimate from Proposition 1.44 and the multinomial formula, leads then finally to   (s) (s) qm (z + ω) − qm (ω)

≤

∞  |m|+k−s  l  + 1  (l + γ)  (k+1)R ω ω k−s+1+|m|

l=1

=

γ=1

 l R  (l + γ)  (k+1)R ω ω k−s+2+|m|

∞  |m|+k−s+1  + l=0

γ=2

54

Chapter 2. Clifford-analytic Eisenstein series for translation groups

Hence, there is a real C > 0 such that    (s)   (s)  qm (z + ω) − qm (ω)  ≤ C R ω∈Ωp \{0}

ω∈Ωp \{0}

1 ω k−s+2+|m|

=CR



ω∈Ωp \{0}

1 ω p+1

which is convergent under the given conditions due to Eisenstein’s lemma (2.8). The cases where |m| = p−k−2+s (under the assumption that p−k−2+s ≥ 0) require already a bit more technical treatment. Again let K ⊂ Ak+1 be an arbitrary compact subset and choose R > 0 so that the closed ball B(0, R) covers completely K. Let z ∈ B(0, R). Now we restrict our consideration, without loss of generality, to those lattice points that satisfy    (k+1)a   0 be sufficiently small so that B(bi , δi )\{bi } contains no poles and no a-points. Then 

ord(f − a; c) = −

c∈P \{b1 ,...,bµ }

where p(f − a; bi ) :=

1

µ 

p(f − a; bi )

(2.24)

i=1



Ak+1

q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)]

∂B(bi ,δi )

which is a finite expression. Proof. The set of poles and a-points are both discrete which follows by the assumption. Therefore, one can choose a period parallelepiped P in such a way that no poles and no a-points lie on its surfaces. Notice that under the given conditions there are at most a finite number of a-points in the interior of each period parallelepiped. A (k + 1)-dimensional parallelepiped has (2k + 2) hypersurfaces. Let us denote them by Bν , Bν , ν = 1, . . . , k + 1 where Bν + ων = Bν . These surfaces should be oriented in such a way that the normal vectors point to the outside. The surfaces Bj are then oriented in the opposite way to Bj . Since P is a period parallelepiped, the surfaces Bi and Bi (i = 1, . . . , k + 1) correspond to each other with respect to the range of values of f . In the sequel we denote the c-points of P by y 1 , . . . , y l . For each j ∈ {1, . . . , l} there exists an εj > 0 and mutually disjoint open balls B(y j , εj ) with ∂P ∩ B(y j , εj ) = ∅ for j = 1, . . . , l.

2.3. Liouville type theorems for generalized elliptic functions l 

Notice that in P \

63

B(y j , εj ) there are no a-points. Now we compute:

j=1



ord(f − a; c) +

c∈P \{b1 ,...,bµ }

= = =

1

Ak+1 1 Ak+1 1 Ak+1

q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)] +

j=1 ∂B(y j ,εj )

,

k+1 

,

q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)]

ν=1 Bν

k+1 

,

, ,

Bν 1 Ak+1

k+1 

−

ν=1

+

 q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)]

, Bν

,

Bν

=

 q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)]

q0 (f (z + ων ) − a)[(Jf )∗ (z)] ∗ [dσ(z)]

ν=1 Bν

+ =

p(f − a; bi )

i=1

∂P

Bν 1 Ak+1

µ 

q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)]

+ =

p(f − a; bi )

i=1

,

l 

µ 

q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)]

 q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)]

0.

In view of the periodicity of f , the differential dσ(f (z)) = [(Jf )∗ (z)] ∗ [dσ(z)] is
invariant under translations of the form z → z + ων . From Theorem 2.9 one may readily deduce (1)

Corollary 2.10. Let f ∈ El (Ωk+1 ) taking only values in Ak+1 . If f has an isolated zero at c ∈ Ak+1 or f (c) = 0, then ord(f ; c + ω) = ord(f ; c)

f or all ω ∈ Ωk+1 .

(2.25)

One can say more: Proposition 2.11. Let f be a non-constant meromorphic function that takes only values in Ak+1 and assume additionally that f has at most isolated poles. Let S stand for the set of poles. Let Ωk+1 be a (k + 1)-dimensional lattice in Ak+1 . We further assume that one can associate with each ω ∈ Ωk+1 a positive real α = α(ω) such that f (z + ω) = α(ω)f (z) (2.26) for all z ∈ Ak+1 \S. If f has an isolated zero at c (or f (c) = 0), then f has an isolated zero at c + ω for all ω ∈ Ωk+1 or f (c + ω) = 0 for all ω ∈ Ωk+1 ,

64

Chapter 2. Clifford-analytic Eisenstein series for translation groups

respectively. In this case, f or all ω ∈ Ωk+1 .

ord(f ; c + ω) = ord(f ; c)

Provided all zeroes of f are isolated,   ord(f ; c) + p(f ; b) = 0 c∈P \S

(2.27)

(2.28)

b∈S

for any period parallelepiped P . Proof. Since α(ω) is a positive real for any arbitrary ω ∈ Ωk+1 , it follows immediately from Proposition 1.30 that c + ω is an isolated zero if and only if c is an isolated zero. Similarly, from f (c) = 0 follows that f (c + ω) = 0 for all ω ∈ Ωk+1 . Next let us consider , q0 (f (z))[(Jf )∗ (z)] ∗ [dσ(z)] Ak+1 ord(f ; c + ω) = ∂B(c+ω,ε)

,

=

q0 (f (z))dσ(f (z))

∂B(c+ω,ε)

=

,

q0 (f (y + ω))dσ(f (y + ω))

∂B(c,ε)

=

,

α(ω) q (f (y))α(ω)k dσ(f (y)) α(ω)k+1 0

∂B(c,ε)

=

,

q0 (f (y))[(Jf )∗ (y)] ∗ [dσ(y)]

∂B(c,ε)

= Ak+1 ord(f ; c) where the parameter ε > 0 has to be chosen sufficiently small. With the same argument one can show (2.28). One simply has to apply the same procedure as in the proof of the third Liouville theorem.
The third Liouville theorem gives thus a first insight in the basic value distribution of the generalized elliptic functions from Clifford analysis that satisfy some special conditions. In particular it revealed that at least under some special conditions there is also a certain balance relation between a-points (a = ∞) and poles in a period cell of a monogenic generalized elliptic function. These statements should be regarded as a promising starting point for more sophisticated versions concerning the cases where manifolds of zeroes and singularities appear. Remarks. As mentioned in [93], generalizations of Theorem 2.8 and Theorem 2.9 can be established in the context of Ak+1 -like linear domain manifolds within the framework of complexified (k + 1)-fold periodic functions satisfying in Ck+1 the complexified Cauchy–Riemann equation. However, additional restrictions have to be put. In the context where the cones of the singularities have only point intersections with the linear domain manifold one can obtain a generalization of

2.4. Series expansions, divisor sums and Dirichlet series

65

Theorem 2.8. This is a consequence of the generalized Cauchy integral formula on IRk -like domain manifolds which was proved in [134]. If the manifolds of c-points have additionally only point intersections with the linear domain manifold, then one obtains also a generalization of Theorem 2.9. These Liouville type theorems indicate a special role of the (k + 1)-fold periodic functions in Ak+1 or its complexification.

2.4

Series expansions, divisor sums and Dirichlet series (1)

(2)

In the classical complex case the function series
n and
n are intimately related to the Riemann zeta function and to Eisenstein series of the type (3). These functions appear explicitly in their Laurent expansion. If 0 < r < 1, then the (1) Laurent expansion of
1 reads exactly ∞

(1)


1 (z) = π cot(πz) =

1  − 2ζ(2n)z 2n−1 , z n=1

and similarly also for m ≥ 1,
(1) m (z)

 2n − 1 1 m = + (−1) 2ζ(2n)z 2n−m . m−1 z 2n≥m

Concerning the second series let us assume without loss of generality that the twodimensional lattice Ω2 has the form Z+Zτ with Im(τ ) > 0. In 0 < r < min{1, |τ |} (2) the Laurent expansion of the associated Weierstraß ℘-function
2 turns out to be ∞  1 (2)
2 (z, Ω2 ) = ℘(z, Ω2 ) = 2 + (2n − 1)G2n (Ω2 )z 2n−2 , z n=2  with Gn (Ω2 ) := (c,d)∈Z×Z\{(0,0)} (cτ +d)−n which converges absolutely for n ∈ IN with n > 2 and does not vanish whenever n is even. The Weierstraß ℘-function serves thus as generating function for the Eisenstein series (3). Next we turn to treat higher dimensional monogenic and s-monogenic analogies. As one can verify by a direct computation the Taylor coefficients of the function (s) |m| ≥ max{0, p − k − 1 + s}
(p) m,s (z) − qm (z), in a sufficiently small neighborhood of the origin have the form  (s) qm (ω),

(2.29)

ω∈Ωp \{0}

involving parameters m and s with |m|+s even and sufficiently large so that (2.29) converges absolutely. This is also true in the more general context of arbitrary real and complex Minkowski type spaces.

66

Chapter 2. Clifford-analytic Eisenstein series for translation groups

Since the series (2.29) with |m| + s even are the Laurent coefficients of the (p) s-meromorphic series
m,s (z), there are actually indices m with |m|+s ≡ 0 mod 2,  (s) for which the associated series (2.29) does not vanish. If ω∈Ωp \{0} qm (ω) van(p)

(s)

ished for all m, then one would obtain that
r,s (z) ≡ qr (z) within a whole ball which is a contradiction. Notice that the series vanish identically whenever (s) |m| + s ≡ 1 mod 2 which is due to the fact that the functions qm are odd in these cases.  (s) In the cases where p ≤ k −s and where s is even, the series ω∈Ωp \{0} q0 (ω) converges and coincides in these cases with the Epstein zeta function  (gtr W tr W g)−s (2.30) ζW tr W (s) = g∈Zp \{0}

where W is the p × (k + 1) matrix which maps the generators of the lattice Lp = Ze0 + Ze1 + · · · + Zep−1 onto the generators of Ωp , i.e., W := (ω1 ω2 . . . ωp ). In the case s = 1 the series  (1) n∈IN qm (n) coincide in the two-dimensional case with the classical Riemann zeta function up to a constant. The series (2.29) are thus generalizations of the classical Riemann zeta function and the Epstein zeta function which appear in a natural way in connection with Clifford-analytic multiperiodic functions. We shall see that these variants of Riemann zeta type functions play a rather central role in all that follows. In order to introduce also a non-vanishing Riemann zeta type function for the cases where |m| + s is odd, we decompose the lattice Ωp := Zω1 + · · · + Zωp into a positive and a negative part (cf. [84]). The positive part of Ωp is defined by Ω+ p

:=

INω1 + Zω2 + Zω3 + · · · + Zωp

∪ ...

INω2 + Zω3 + · · · + Zωp

∪

INωp .

Consequently, the negative part of the lattice Ωp is defined by   + Ω− p := Ωp \{0} \Ωp so that

− − and Ω+ z ∈ Ω+ p ⇔ −z ∈ Ωp p ∪ Ωp ∪ {0} = Ωp .

In particular, for k = 1 and ω1 = 1 one has: Ω+ 1 = IN Now we can introduce:

Ω− 1 = −IN Ω1 = IN ∪ −IN ∪ {0} = Z.

2.4. Series expansions, divisor sums and Dirichlet series

67

Definition 2.12 (Generalized Riemann zeta function of Clifford analysis in Ak+1 ). be a multiLet p, s ∈ IN with 1 ≤ p ≤ k + 1 and 1 ≤ s ≤ k. Let further l ∈ INk+1 0 index with |l| ≥ max{0, p − k + s}. Then the generalized Riemann zeta function of Clifford analysis in Ak+1 is defined by  (s) Ω ql (ω). (2.31) ζMp (l, s) := ω∈Ω+ p

To prove the convergence one applies first the estimate (1.50) that we devel(s) oped for the functions ql both for s odd and s even and after that Eisenstein’s lemma. This shows that the series (2.31) converge indeed absolutely under the given conditions. In the cases where we have additionally |l| + s ≡ 0 mod 2 we obtain the relation  Ω (s) ql (ω). 2ζMp (l, s) = ω∈Ωp \{0} (s)

From the estimates on the functions qn we can readily derive that |l|−1 Ω

ζMp (l, s) ≤

-

(k + 1 − s + µ) ζW tr W

µ=0

1 2

 (k + 1 − s + |l|)

where ζW tr W is again the Epstein zeta function associated to the matrix W tr W . The closed representation formulas that we developed in the second part of Sec(s) tion 1.6 for the functions ql enable us next to describe the generalized Riemann zeta functions explicitly in terms of finite sums of variants of Dirichlet series with polynomial coefficients of the form  P (g)(gtr S g)−t . (2.32) δ(P ( · ), S, t) := g∈Zp \{0}

Here P denotes a real-valued polynomial in g1 , . . . , gp , S a positive definite matrix and t a real or complex parameter. Notice that a series of the form (2.32) is convergent if Re(t) − deg(P ) > (p/2). If P ≡ 1, then δ(1, S, t) coincides with the Epstein zeta function associated to the matrix S, i.e., ζS (t). We also consider the positive part of the Dirichlet series, which is defined by  P (g)(gtr S g)−t . (2.33) D(P ( · ), S, t) := g∈Zp+

D(P ( · ), S, t) provides a canonical generalization of the classical Dirichlet series  D(P ( · ), t) := P (n)n−t n∈IN

when putting p = 1 and S = (1). We observe that δ(P ( · ), S, t) = 2D(P ( · ), S, t) if P is an even function and that δ(P ( · ), S, t) = 0 if P is odd.

68

Chapter 2. Clifford-analytic Eisenstein series for translation groups

In the case where s is an even positive integer, we get directly by Theorem 1.49 that  Ω D(a(k, n, p)ω n−2p , W tr W, k − s + 1 + 2|n| − 2|p|) ζMp (n, s) = 0≤2p≤n

with a(k, n, p) = (−1)|n|−|p|

k − s 2

|n|−|p| (n

n! . − 2p)!p!

(2.34)

Notice, ω n−2p means ω0n0 −2p0 · · · ωknk −2pk . In the case where |n| is even we have furthermore 1  Ω ζMp (n, s) = δ(a(k, n, p)ω n−2p , W tr W, k − s + 1 + 2|n| − 2|p|). 2 0≤2p≤n

In the case where s is odd with s < k we can use Theorem 1.50. Due to the more (s) complicated structure of the functions qn for these cases, we will get slightly more complicated representation formulas for the associated generalized Riemann zeta type functions in terms of these variants of Dirichlet series. To proceed in this direction we first observe that one can write ω n−2p

ω k−s+2|n|−2|p|

k k   nj − 2pj ω n−2p−τ (j) · ej = (nj − 2pj ) ej ωj

ω k−s+2|n|−2|p| j=0 j=0

(2.35)

and also  ω n−2n+τ (0) 1 ω n−2n+τ (j) = − ej .

ω k−s+2|n|−2|p| ω

ω k−s+2+2|n|−2|p| j=1 ω k−s+2+2|n|−2|p| k

ω n−2p

(2.36)

Inserting these relations into the representation formula (1.65) leads to qn(s) (ω) 1 = k−s

 0≤2p≤n



.

a(k, n, p) · 

ω n−2p+τ (0)

ω k−s+2+2|n|−2|p|

−

ω n−2p+τ (j)

j=1

ω k−s+2+2|n|−2|p|

3 ω + bj (k, n, p) ej ,

ω k−s+2|n|−2|p| j=0 k 



k 

ej 

n−2p−τ (j)

(2.37) after having put bj (k, n, p) = (−1)|n|−|p|

k − s 2

|n|−|p|

(nj − 2pj ) j = 0, 1, . . . , k.

For the sake of readability let us further introduce the real-valued polynomials Aj (ω) := ω n−2p+τ (j)

Bj (ω) := ω n−2p−τ (j)

(2.38)

2.4. Series expansions, divisor sums and Dirichlet series

69

so that we arrive at the compact formula Ω

ζMp (n, s) .   a(k, n, p)A0 (ω)  1 b0 (k, n, p)B0 (ω)  = + k−s

ω k−s+2+2|n|−2|p|

ω k−s+2|n|−2|p| + 0≤2p≤n +

k  j=1

4

ω∈Ωp

5 3  bj (k, n, p)Bj (ω)  a(k, n, p)Aj (ω) − ej .

ω k−s+2|n|−2|p|

ω k−s+2+2|n|−2|p| + +

ω∈Ωp

ω∈Ωp

From this formula we finally obtain also for the cases s < k where s is odd the following explicit representation formula for paravector components of the generalized Riemann zeta function in terms of finite sums of real-valued Dirichlet series with explicitly determined polynomial coefficients:   1 Ω D(a(k, n, p)A0 (W · ), W tr W, k − s + 2 + 2|n| − 2|p|) ζMp (n, s) = k−s 0≤2p≤n

+D(b0 (k, n, p)B0 (W · )W tr W, k − s + 2|n| − 2|p|) k   D(bj (k, n, p)Bj (W · ), W tr W, k − s + 2|n| − 2|p|) + j=1

−D(a(k, n, p)Aj (W · ), W tr W, k − s + 2 + 2|n| − 2|p|)ej

 .

In the case |n| ≡ 1 mod 2 one obtains   1 Ω δ(a(k, n, p)A0 (W · ), W tr W, k − s + 2 + 2|n| − 2|p|) ζMp (n) = 2(k − s) 0≤2p≤n

+δ(b0 (k, n, p)B0 (W · )W tr W, k − s + 2|n| − 2|p|) k   + δ(bj (k, n, p)Bj (W · ), W tr W, k − s + 2|n| − 2|p|) j=1

−δ(a(k, n, p)Aj (W · ), W tr W, k − s + 2 + 2|n| − 2|p|)ej

 .

Next we turn to generalizations of the Eisenstein series (3) in this framework. Suppose that Ωk+1 = Zω0 + · · · + Zωk is an arbitrary (k + 1)-dimensional lattice in Ak+1 . By a rotation, we can transform this lattice into a special lattice of the form Ω∗k+1 = Zτ + Ωk where Ωk is completely contained in IRk , i.e., Sc(Ωk ) = 0 and where Sc(τ ) > 0. The associated Riemann zeta functions  1 Ω∗ (s) ζMk+1 (m, s) = qm (ατ + ω) (2.39) 2 (α,ω)∈Z×Ωk \{(0,0)}

70

Chapter 2. Clifford-analytic Eisenstein series for translation groups

are precisely the Laurent coefficients of the s-monogenic generalized Weierstraß functions associated to Ω∗k+1 . They generalize to the higher dimensional case the  expression (c,d)∈Z×Z\{(0,0)} (cτ +d)−n . Regarding τ as a hypercomplex variable of the half-space H + (Ak+1 ) leads then to s-monogenic generalizations of the classical Eisenstein series of type (3) to Clifford analysis in this sense. More generally as in our previous works [84, 86, 88] we introduce here: Definition 2.13. Let p ∈ {1, . . . , k} and let Ωp = Zω1 +· · ·+Zωp be a p-dimensional lattice in Ak+1 which is contained in spanIR {e1 , . . . , ek }. Let s < k + 1 be a with |m| + s ≡ 0 mod 2 and |m| ≥ positive integer. For a multi-index m ∈ INk+1 0 max{0, p + s + 1 − k} we introduce the following type of s-monogenic Eisenstein series on the right half-space:  (s) G(p) qm (αz + ω) z ∈ H r (Ak+1 ). (2.40) m,s (z) := (α,ω)∈Z×Ωp \{(0,0)}

For the monogenic case s = 1 we introduced these types of Eisenstein series in [84] for p = k. The convergence proof from [84, 86] can easily be adapted to the more general setting by using the more general estimate for the s-monogenic (s) functions qn (z) from Proposition 1.44. The classical Eisenstein series (3) have a very interesting Fourier expansion, from the number theoretic point of view. It is directly related to the Riemann zeta function and representation numbers of sums of divisors [49]. (p)

In what follows let us abbreviate the 1-monogenic Eisenstein series Gm,1 (p)

by Gm for simplicity. In [84, 86] we determined the Fourier expansion of the (k) 1-monogenic Eisenstein series Gm where we considered without qualitative loss of generality the orthonormal lattice Lk = Ze1 + · · · + Zek and, in view of the Cauchy–Riemann system, multi-indices with m0 = 0. Theorem 2.14 (Fourier expansion). Let m = (0, m1 , . . . , mk ) ∈ INk+1 be a multi0 (k) index with |m| ≡ 1 mod 2, |m| ≥ 3. Then the Eisenstein series Gm,1 associated with the orthonormal lattice Lk in IRk have a Fourier expansion on the right halfspace,  s 2πi s,x −2π s x0 Lk |m| )e G(k) σm (s)(1 + i e m (z) = 2ζM (m) + Ak+1 (2πi)

s k s∈Z \{0}

(2.41) where σm (s) =



rm

(2.42)

r|s

and where r|s means that there is an α ∈ IN such that αr = s. For the detailed proof we refer to [84, 86]. The basic idea of the proof is to (k) expand first the subseries
m,1 (z) (|m| ≥ 2) on H r (Ak+1 ) into a Fourier series of

2.4. Series expansions, divisor sums and Dirichlet series the form



71

αf (r, x0 )e2πi r,x .

r∈Zk

By a direct computation one obtains    αf (r, x0 ) = qm (z + m) dx1 · · · dxk = 0. m∈Zk

[0,1]k

For r = 0 one applies the partial integration method successively (integrating qm up until we obtain q0 and differentiating the exponential terms). This leads finally to  αf (r, x0 ) = (2πi)|m| rm

q0 (z)e−2πi r,x dx1 · · · dxk .

IRk

The value of the remaining integral is well known. See for example [154, 108, 23]. It can be evaluated by applying the residue theorem for monogenic functions,  Ak+1

r  −2π r x0 q0 (z)e−2πi r,x dx1 · · · dxk = 1+i e 2

r IRk

so that one finally obtains for x0 > 0 that
(k) m (z) =

Ak+1 2



r  −2π r x0 2πi r,x e (2πi)|m| rm 1 + i e .

r

(2.43)

r∈Zk \{0}

(k)

Rearrangement arguments allow us next to rewrite Gm (z) in the form G(k) m (z)

= 2ζ

Lk

(m) + 2

∞  


m (αz)

(2.44)

α=1 m∈Zk

where we used the notation introduced above. Applying (2.43) to (2.44) yields Gm (z) = 2ζ Lk (m) + 2(2πi)|m| (k)



Ak+1 2

  ∞  α=1 r∈Zk

 αr e2πi αr,x e−2π αr x0 rm 1 + i αr

which finally may be expressed in the form (2.41). Here we have indeed a nice analogy between the form of the Fourier expansion of the classical Eisenstein series (3) and the structure of the Fourier expansion of the higher dimensional variant defined in (2.40). The ordinary Riemann zeta function is replaced in the higher dimensional monogenic context by the generalized paravector valued variant (2.31). As we have seen previously, their paravector

72

Chapter 2. Clifford-analytic Eisenstein series for translation groups

component can in turn be expressed in terms of a finite sum of scalar-valued Dirichlet series with polynomial coefficients, generalizing the Epstein zeta function. The Fourier coefficients αf (r) for r = 0 of the complex Eisenstein series σm (s) =



rm−1

(2.45)

r|s

where r|s means that there is an α ∈ IN such that αr = l are thus generalized by the expression (2.42) which can further be expressed in terms of (2.45): σm (s) = sm σ−|m| (gcd(s1 , . . . , sm−1 )).

(2.46)

As a consequence of the monogenicity, the monogenic plane wave function appears as a natural generalization of the classical exponential function in the sense of Cauchy–Kowalewski extension. Notice further that infinitely many Fourier coef(k) ficients do not vanish. Hence the series Gm (z) are definitely non-trivial functions whenever |m| is odd. In Chapter 3 we will observe furthermore, that they provide elementary building blocks for the generation of families of Clifford-analytic modular forms for larger discrete groups, in particular, also for the special hypercomplex modular groups Γp . This in turn provides us with an analogy to the classical complex case. (k) However, the proper series Gm (z) are not yet modular forms for Γk . Only their set of singularities Qe1 + · · · + Qek is totally invariant under the action of Γk . This in fact provides a difference from the complex case.

2.5

The integer multiplication of the Clifford-analytic Eisenstein series

One classical result from complex analysis (cf. e.g., [130]) states that the classical cotangent function π cot(πz) is completely characterized by the property of being 1 at each point m ∈ Z an odd and meromorphic function with principal parts z−m that satisfies additionally the duplication formula 1 2f (2z) = f (z) + f (z + ). 2

(2.47)

The doubly periodic Weierstraß ℘-function also satisfies an analogous duplication formula (cf. e.g., [82]):  v (2.48) 4℘(2z) = ℘(z + ). 2 v∈V(2)

2.5. The integer multiplication of the Clifford-analytic Eisenstein series

73

Here in the summation, the symbol V(2) stands again for the canonical system of representatives in Ω/2Ω, as introduced in Definition 2.3. Conversely, every mero1 morphic function with principal parts (z−ω) 2 at each ω ∈ Ω that satisfies a functional equation of type (2.48) coincides with the ℘-function up to a constant. From (2.48) one can further directly derive that the sum of the integer division values gives  ℘(v/2) = 0. 0=v∈V(2)

In [84] and [85] we deduced analogous duplication formulas for the monogenic p-fold periodic generalizations of the cotangent and the monogenic (k + 1)-fold periodic Weierstraß ℘τ (i) -functions and showed that these generalized duplication formulas permit an analogous characterization of these functions as in the complex case. In our recent paper [92] we developed more general multiplication formulas for a larger class of s-monogenic translative Eisenstein series. In this section and the following two we present the results from our recent paper [92]. In this section and in Section 2.6 we describe the multiplication of the smonogenic Eisenstein series with general integer multiplicators. In Section 2.7 we (p) will treat the hypercomplex multiplication of the series
m,s . Throughout this section and the next one Ωp = Zω1 + · · · + Zωp stands always for an arbitrary p-dimensional lattice with 1 ≤ p ≤ k + 1 in Ak+1 . To study the integer multiplication of the s-monogenic Eisenstein series means to describe the relationship between the monogenic and polymonogenic Eisenstein series of an arbitrary lattice Ωp and those of a p-dimensional sublattice Ω∗p ⊂ Ωp that has the form Ω∗p = nΩp with a positive integer n ≥ 2. By Vp (n) := {m1 ω1 + · · · + mp ωp ; m1 , . . . , mp ∈ IN0 , 0 ≤ m1 , . . . , mp < n} we denote the canonical system of representatives of Ωp /nΩp . For the cases where m ∈ INk+1 is a multi-index with |m| ≥ max{0, p − k + s} 0 (p) we can immediately infer by a direct rearrangement that the series
m,s satisfy the following integer multiplication formulas: nk+1+|m|−s
(p) m,s (nz) =

 v∈Vp (n)


(p) m,s (z + v/n).

(2.49)

74

Chapter 2. Clifford-analytic Eisenstein series for translation groups

Simply consider, 

k+1+|m|−s (p)
(p)
m,s (nz) m,s (z + v/n) − n

v∈Vp (n)



=





(s) qm (z + v/n + ω) − nk+1+|m|−s

v∈Vp (n) ω∈Ωp



k+1+|m|−s

= n



(s) qm (nz + ω)

ω∈Ωp (s) qm (nz

+ v + nω)

v∈Vp (n) ω∈Ωp

− nk+1+|m|−s



(s) qm (nz + ω)

ω∈Ωp



k+1+|m|−s

= n



(s) qm (nz + v + ω  )

v∈Vp (n) ω  ∈nΩp



− n

k+1+|m|−s



(s) qm (nz + v + ω  ) = 0,

v∈Vp (n) ω  ∈nΩp (s)

where we used furthermore that qm is IR-homogeneous of degree s − |m| − k − 1. The following two statements permit us to establish an analogous statement is a multi-index with for the cases with p − k − 1 + s ≥ 0 where m ∈ INk+1 0 |m| = p − k − 1 + s and where s + |m| is odd. Proposition q2.15. Let Ωp = Zω1 + · · · + Zωp ⊂ Ak+1 be a p-dimensional lattice.  Let v = i=1 αi ωi with 0 = αi < n where 1 ≤ q ≤ p. Write further v = q k+1 be a multi-index i=1 (n − αi )ωi . Let s ∈ IN with s < k + 1 and let m ∈ IN0 such that |m| + s is odd and that furthermore |m| ≥ p − k − 1 + s. Then       (s) (s) (s) (s) qm qm (nω + v) − qm (nω) + (nω + v  ) − qm (nω) ω∈Ωp \{0} ω∈Ωp \{0} (2.50) = −qm (v) − qm (v  ). (s)

(s)

(s)

Proof. Both series converge absolutely. The functions qm are odd, under the condition that |m| + s is odd. Hence we can rewrite the left-hand side of (2.50) in the way  q q q   (s)    (s) qm (nω + αi ωi ) − qm (n(ω − ωi ) + αi ωi ) . ω∈Ωp \{0}

i=1

i=1

i=1

This series in turn can be split into the following two parts:  q q q q   (s)     (s) ωi + αi ωi ) − qm (n(r − 1) ωi + αi ωi ) (2.51) qm (nr r∈Z\{0}

+



 ω∈Ωp \Z(

q  i=1

i=1 (s)

qm (nω + ωi )

i=1 q  i=1

i=1 (s)

αi ωi ) − qm (n(ω −

q  i=1

ωi ) +

i=1 q  i=1

 αi ωi ) . (2.52)

2.5. The integer multiplication of the Clifford-analytic Eisenstein series

75

Here, the prime behind Ωp means that the origin is omitted in the summation. q (s) From lim qm (z) = 0 follows that (2.52) equals zero. ω ∈ Ωp \Z( i=1 ωi ) implies z→∞ q q  (s) namely that also ω − i=1 ωi ∈ Ωp \Z( ωi ). In view of the oddness of qm , the i=1

sum in (2.51) simplifies to q q q q   (s)     (s) qm (nr ωi + αi ωi ) − qm (n(r − 1) ωi + αi ωi ) r∈IN (s)

i=1 q 

−qm (nr =

i=1 q 

ωi −

(s)

i=1 q 

αi ωi ) + qm (n(r + 1)

i=1 i=1 q q (s)  (s)  −qm ( αi ωi ) − qm ( (n i=1 i=1

i=1 q 

ωi −

i=1

 αi ωi )

i=1

− αi )ωi ).




This proposition permits us to establish next Lemma 2.16. Let Ωp := Zω1 + · · · + Zωp be a p-dimensional lattice in Ak+1 where 1 ≤ p ≤ k + 1. Let further Vp (n) be the canonical system of representatives of be Ωp /nΩp where n ≥ 2 is an integer. Let s ∈ IN with s < k + 1 and let m ∈ INk+1 0 a multi-index such that |m| + s is odd and that furthermore |m| ≥ p − k − 1 + s. Then    (s) (s) (s) [qm (nω + v) − qm (nω)] = − qm (v). (2.53) v∈Vp (n)\{0} ω∈Ωp \{0}

v∈Vp (n)\{0}

Remark. Lemma 2.16 provides us with a generalization of the particular relation    (1) (1) (1) q0 (2ω + v) − q0 (2ω) = −q0 (v), (2.54) ω∈Ωk \{0}

which we established earlier in [84, 85] for proving the periodicity of the particular (k) function
0,1 . In equation (2.54) v is supposed to stand for an arbitrary element from Vk (2)\{0}. Note that (2.54) is not satisfied if one replaces 2 by an arbitrary different integer n > 2. Lemma 2.16 permits us next to establish be Theorem 2.17. Let p, s ≤ k and assume that p − k − 1 + s ≥ 0. Let m ∈ INk+1 0 a multi-index with |m| = p − k − 1 + s. Suppose further that s + |m| is odd. Then (p) (2.49) is also satisfied by the associated series
m,s . Remark. In the monogenic case this theorem deals exclusively with the configuration p = k and m = 0. (p)

Proof of Theorem 2.17. In the context considered here the series
m,s has the more complicated form   (s) (p) (s) (s) (2.55) qm (z + ω) − qm (ω)
m,s (z; Ωp ) = qm (z) + ω∈Ωp \{0}

76

Chapter 2. Clifford-analytic Eisenstein series for translation groups

in order to have convergence. In the 1-monogenic case we are dealing with the particular configuration p = k and m = 0. The expression 

(p)

(p)


m,s (z + v/n) − nk+1+|m|−s
m,s (nz)

(2.56)

v∈Vp (n)

can hence be rewritten in the form   (s) qm (z + v/n) +



(s)

(s)

[qm (z + ω + v/n) − qm (ω)]

v∈Vp (n) ω∈Ωp \{0}

v∈Vp (n) (s) −nk+1+|m|−s qm (nz)



= nk+1+|m|−s

− nk+1+|m|−s

+n





(s)

(s)

[qm (nz + nω + v) − qm (nω)]

v∈Vp (n) ω∈Ωp \{0}

− nk+1+|m|−s



+n



(s)

(s)

[qm (nz + ω) − qm (ω)]

ω∈Ωp \{0}

(s)

qm (nz + v)

v∈Vp (n)\{0} k+1+|m|−s

(s)

(s)

(s) −nk+1+|m|−s qm (nz)

= nk+1+|m|−s

(s)

[qm (nz + ω) − qm (ω)]

ω∈Ωp \{0}

qm (nz + v)

v∈Vp (n) k+1+|m|−s







(s)

(s)

[qm (nz + nω + v) − qm (nω)]

v∈Vp (n) ω∈Ωp \{0}

−n

k+1+|m|−s



(s)



−nk+1+|m|−s

(s)

[qm (nz + nω) − qm (nω)]

ω∈Ωp \{0}



(s)

(s)

[qm (nz + nω + v) − qm (nω + v)]

v∈Vp (n)\{0} ω∈Ωp



k+1+|m|−s

= n

(s)

qm (nz + v)

v∈Vp (n)\{0}



k+1+|m|−s

+n



(s)

(s)

[qm (nz + nω + v) − qm (nω)]

v∈Vp (n)\{0} ω∈Ωp \{0}



−nk+1+|m|−s

v∈Vp (n)\{0}

−nk+1+|m|−s = nk+1+|m|−s





(s)

(s)

[qm (nz + v) − qm (v)] 

(s)

(s)

[qm (nz + nω + v) − qm (nω + v)]

v∈Vp \{0} ω∈Ωp \{0}

 v∈Vp (n)\{0}

(s)

qm (v) +





 (s) (s) (qm (nω + v) − qm (nω)) .

v∈Vp (n)\{0} ω∈Ωp \{0}

(2.57) Since |m| + s is odd, Lemma 2.16 can be applied to (2.57). This allows us to establish the multiplication formula.
Remark. Let us assume that p ≤ k + 1, s ≤ k, p − k − 1 + s ≥ 0 and that m is a multi-index from INk+1 with |m| = p − k − 1 + s. Whenever s + |m| is even, 0

2.5. The integer multiplication of the Clifford-analytic Eisenstein series

77

we cannot apply Lemma 2.16 in the previous line of the proof of Theorem 2.17. Nevertheless, the modulus of the expression within the brackets of (2.57) must be finite. This allows us to conclude at least that also in these cases there is a paravector constant C ∈ Ak+1 with 
(p) (2.58) nk+1+|m|−s
(p) m,s (nz) = m,s (z + v/n) + C. v∈Vp (n)

In the special case where p = k + 1 and where m is a multi-index from INk+1 0 with |m| = s, we are dealing with s-monogenic generalizations of the Weierstraß ℘-function. In this special context one can use a very elegant method, involving the s-monogenic generalizations of the Weierstraß ζ-function, in order to show that C = 0. What follows is an extension in several directions of R. Fueter’s calculations and methods that were presented in [63] pp. 236–245 exclusively in the context of the particular problem of the multiplication of the four-fold periodic quaternionic 1-monogenic ℘-function. We show more generally: Theorem 2.18. Let Ωk+1 be a (k + 1)-dimensional lattice in Ak+1 . Let n ≥ 2 be a positive integer and let Vk+1 (n) be the canonical system of representatives of Ωk+1 /nΩk+1 . Then the s-monogenic generalizations of the Weierstraß ℘-function satisfy all the integer multiplication formulae  nk+1
(k+1) (nz) =
(k+1) (z + v/n) (2.59) s,s s,s v∈Vk+1 (n)

where s is a multi-index of length |s| = s. Proof. Let us take an arbitrary (k+1)-fold periodic s-monogenic translative Eisen(k+1) stein series
s,s with |s| = s. As a consequence of the simple fact that |s| ≥ 1 there is an i ∈ {0, . . . , k} so that si > 0. Furthermore, there is a multi-index r ∈ IN0 k+1 of length s − 1 with r + τ (i) = s. Let us now take the k + 1many s-monogenic Eisenstein series associated to the multi-indices r + τ (j) where (k+1) j = 0, 1, . . . , k. The original Eisenstein series
s,s is among them. In view of (k+1) |r| ≥ 0, all the Eisenstein series
r+τ (j),s j = 0, 1, . . . , k can be generated by partial differentiation from one global s-monogenic quasi-periodic primitive, namely the s-monogenic Weiertraß ζ-functions ζr,s (see Section 2.1). From (2.58) follows that all the k + 1 partial derivatives of  (k+1) (k+1) Ej (z) := nk+1
r+τ (j),s (nz) −
r+τ (j),s (z + v/n) (2.60) v∈Vk+1 (n)

vanish identically. Therefore, E0 , E1 , . . . , Ek must be constants. This statement could be established alternatively by applying generalizations of the classical Liouville’s theorem to the class of s-monogenic functions, cf. [11, 105]. After this a comparison of the Laurent coefficients provides us with this statement.

78

Chapter 2. Clifford-analytic Eisenstein series for translation groups (k+1)

Since all the functions
r+τ (j),s can be generated by partial differentiation from the function ζr,s , one may now integrate (2.60), so that we get 

nk ζr,s (nz) =

ζr,s (z + v/n) +

k 

Ej xj + E,

(2.61)

j=0

v∈Vk+1 (n)

where E denotes a further independent Clifford constant. If we next apply on both sides of (2.61) a shift of the argument of the form z → z + ωh (h = 0, . . . , k), then we arrive at

 [s] nk ζr,s (nz)+nηh =





[s]

ζr,s (z+v/n)+ηh

k   + Ej (xj +ωhj )+E. (2.62) j=0

v∈Vk+1 (n)

k

Here, and in the sequel we use the notation ωh = j=0 ωhj ej . In view of (2.61), (2.62) implies k  

 [s] k+1 [s] ηh = ηh + Ej ωhj . (2.63) n j=1

v∈Vk+1 (n) k+1

Since Vk+1 (n) = n

, the equation (2.63) simplifies to k 

Ej ωhj = 0.

(2.64)

j=1

Let us write further



ω00  ω10 det W = det   ··· ωk0

··· ··· ···

 ω0k ω1k   ···  ωkk

(2.65)

and Θhj for the adjoint determinant associated with the element ωhj . Then k 

ωhi Θhl = δil · det W.

(2.66)

h=0

From (2.64) follows in particular that k  h=0

Θhi

k



 Ej ωhj

= 0.

(2.67)

j=1

Applying (2.66) to (2.67) leads to the system Ei det W = 0 for all i = 0, . . . , k. The k + 1 primitive periods ωh (h = 0, . . . , k) of the (k + 1)-fold periodic series (k+1)
r+τ (j),s are all linearly independent. For this reason det W = 0. Hence, Ej = 0 for all j = 0, 1, . . . , k. The integer multiplication formulas for the s-monogenic generalized ℘-functions are hereby established.


2.6. Characterization theorems

79

For the remaining cases where p, s, m are chosen so that s + |m| is an even positive integer where |m| = p − k − 1 + s with p < k + 1, this argument cannot be adapted so directly. In order to show that Ej = 0 for all j = 0, . . . , k we used that the period matrix is an invertible matrix having full rank k + 1. Theorem 2.18 permits us now directly to show that the sum of the integer division values of the s-monogenic generalizations of the Weierstraß ℘-function vanishes. We only have to apply the following limit argument:  
(k+1) (v/n) = lim
(k+1) (z + v/n) s,s s,s z→0

v∈Vp (n)\{0}

=

v∈Vp (n)\{0}

  lim nk+1+|m|−s
(k+1) (nz) −
(k+1) (z) = 0. s,s s,s

z→0

The limit calculus in the line above is allowed since all expressions that are involved are s-monogenic in a sufficiently small annular domain around zero. The limit value vanishes since the Laurent expansion of  (z) = qs(s) (z) + qs(s) (z + ω) − qs(s) (ω)
(k+1) s,s ω∈Ωk+1 \{0} (s)

has the form qs (z) + O(z) in a neighborhood around the origin. This result provides a generalization of the classical result that the sum over the integer division values of the holomorphic Weierstraß ℘-function vanishes (see e.g., [82] p. 83, [103]). One can say more. Whenever |m| + s is an even positive integer so that  (s) qm (z + ω) ω∈Ωp

converges, then the sum over the integer division values equals the value of one of those generalized Riemann zeta functions defined in Section 2.4, namely  (s) 2ζ Ωp (m, s) = qm (ω). ω∈Ωp \{0} (p)

If |m| + s is odd, then the sum over the integer division values of the series
m,s (p) gives zero, which is a consequence of the oddness of
m,s in the cases where |m| + s is odd.

2.6

Characterization theorems

For a number of choices of parameters p, m, s we have shown in the previous section (p) that the s-monogenic Eisenstein series
m,s are solutions of a functional equation of the form (2.49). In this section we conversely want to study now the general class of s-monogenic Clifford-valued functions that solve such kinds of functional

80

Chapter 2. Clifford-analytic Eisenstein series for translation groups

equations. In particular we are interested in understanding under which additional conditions one obtains a characterization of the class of s-monogenic Eisenstein series in terms of these types of functional equations. In our recent paper [92] we established two theorems in this direction. First we proved: Theorem 2.19. Let s < k + 1, p ≤ k + 1 and let Ωp = Zω1 + · · · + Zωp be be a multi-index with |m| ≥ a p-dimensional lattice in Ak+1 . Let m ∈ INk+1 0 max{0, p − k − 1 + s}. Assume further that g : Ak+1 → Cl0k is a left or right (s) s-monogenic function with principal parts qm (z − ω) at each ω ∈ Ωp . If g satisfies  nk+1+|m|−s g(nz) = g(z + v/n), (2.68) v∈Vp (n) (p)

then there is a C ∈ Cl0k such that g(z) =
m,s (z) + C for all z ∈ Ak+1 \Ωk+1 . (p)

Proof. To show the assertion consider h(z) = g(z) −
m,s (z) which is a left or right s-monogenic function, respectively, in the whole space Ak+1 . For all configurations with |m| = s with |m| ≥ max{0, p − k + s}, as well as for multi-indices m ∈ INk+1 0 in the case p = k + 1, and also for all configurations with p − k − 1 + s ≥ 0 where m is a multi-index with |m| = p − k − 1 + s where |m| + s is additionally odd, we have managed to establish the integer multiplication formulas for the function (p) series
m,s in the previous section. For all these cases we know then for sure that also the function h satisfies  nk+1+|m|−s h(nz) = h(z + v/n), h(0) = h0 . (2.69) v∈Vp (n)

Let us now first concentrate on these cases and let us assume that h ≡ h0 . Further, put β := ω1 + · · · + ωp . Since the maximum principle holds for the class of smonogenic functions, we may conclude that there is an element c ∈ ∂B(0, nβ) such that

h(z) < h(c) for all z ∈ B(0, nβ). In view of

p    n−1   β < 2β αi ωi )/n ≤ β + (c + n i=1

for all 0 ≤ αi < n, we obtain nk+1+|m|−s h(c)



=

v∈Vp (n)


β. Suppose that f : B(0, r) → Cl0k is a left (right) s-monogenic function in the whole ball B(0, r) that has a continuous extension to ∂B(0, r) and that there is furthermore another C |m| -function χ : B(0, r) → Cl0k (necessarily s-monogenic in B(0, r)) such that for a fixed real δ > 0 holds  nδ f (nz) = f (z + v/n) + χ(z), (2.70) v∈Vp (n) ◦

whenever the points z, nz, z + v/n for all v ∈ Vp (n) lie in the ball B (0, r). If χ satisfies the growth condition M (χm , r) < (nδ+|m| − np )M (fm , r),

(2.71)

82

Chapter 2. Clifford-analytic Eisenstein series for translation groups

then f must be an s-monogenic polynomial of a total degree that is less than or equal to |m|. Proof. Let us take an arbitrary left s-monogenic function f that satisfies (2.70). |m| If one applies ∂∂zm on both sides of equation (2.70), then one gets nδ+|m| fm (nz) =



fm (z + v/n) + χm (z).

(2.72)

v∈Vp (n)

Let t be a positive real number within β < t < r. If z ∈ B(0, t), then (z + v)/n ∈ B(0, t) for all v ∈ Vp (n). If we replace z by z/n in (2.72), then we obtain       nδ+|m| fm (z) ≤ fm ((z + v)/n) + χm (z/n) v∈Vp (n)

≤ np M (fm , t) + M (χm , t) < np M (fm , t) + nδ+|m| M (fm , t) − np M (fm , t) = nδ+|m| M (fm , t). In view of M (fm , t) < M (fm , t), we have thus arrived at a contradiction. Therewith length |m|. Hence f must be an s-monogenic fore, fr = 0 for all r ∈ INk+1 0 polynomial of a total degree being less than or equal to |m|.
This theorem generalizes a result from our earlier paper [87] in which a special version of this lemma was proved exclusively for the monogenic case, and, exclusively for the configuration p ≤ k, m = 0, n = 2 and χ ≡ 0. Now we can draw the following conclusion: A non-constant s-monogenic function that satisfies a functional equation of the form (2.49) must have singularities. Whenever the singularities are all isolated and distributed in the form of a lattice and have furthermore all the same order and principal parts, then this function (p) coincides at least up to a constant with
m,s . A question that arises in a natural way in this context is if there are also non-constant functions that satisfy these multiplication formulas which do not (p) coincide with
m,s . Let us finish this section by giving one example. For a multik+1 with |m| ≥ max{0, p + s + 1 − k} consider the s-monogenic index m ∈ IN0 functions  (s) qm (αz + ω) Sc(z) > 0 G(p) m,s (z) = (α,ω)∈Z×Ωp \{(0,0)}

where Ωp is a p-dimensional lattice in spanIR {e1 , . . . , ek }, as introduced in Section 2.4. Since the series is normally convergent one can directly conclude by rearrangement arguments in order to show that  nk+1−s|m| G(p) G(p) m,s (nz) = m,s (z + v/n). v∈Vp (n)

2.7. Lattices with hypercomplex multiplication

83

Notice that these functions are only s-monogenic in the right half-space H r (Ak+1 ). They have non-isolated singularities in each point of the dense set Qω1 + · · · + Qωp and are thus not meromorphic in the whole space Ak+1 .

2.7

Lattices with hypercomplex multiplication

In the previous two sections we discussed integer multiplication for the s-monogenic Eisenstein series. The treatment could be performed for all arbitrary lattices in Ak+1 without putting any restrictions on the generators of the lattice. This was due to the fact that we have nΩ ⊂ Ω for all n ∈ Z, independently from the form of the lattice. In this section we present more general multiplication formulas for the s-monogenic Eisenstein series that involve more generally hypercomplex multipliers from Ak+1 , in particular, paravector multipliers stemming from the proper lattice. However, in order to meet this end, it will be necessary to put number theoretical conditions on the generators of the lattice. Also the results from this section stem from our recent paper [92]. Before we go into detail let us first recall the basic results for the classical complex case and give some historical background information on this problem. Without loss of generality let us assume as usual that a √ complex lattice Ω has the form Ω = Z + Zτ with Im(τ ) > 0. Whenever τ ∈ Z[ −D] with a positive square-free integer D, then the relation λΩ ⊂ Ω is also satisfied for some complex numbers from C\Z, namely exactly by all the elements of the lattice and no more. This special property has been known for quite a long time. For lattices with this property the terminology lattices with complex multiplication has been established. For more details about the classical theory, see for instance [82, 103]. Now suppose that f (z) is an elliptic function associated with such a lattice. Then fλ (z) := f (λz) is again an elliptic function associated with the same lattice for all λ ∈ Ω. As a consequence of this one obtains an explicit connection between fλ and f in terms of a functional equation. In particular one gets a rather simple functional equation between ℘λ and ℘ which provides a generalization of (2.48) for complex multipliers. From this particular functional equation one can deduce an explicit formula for the trace of the complex division values of the Weierstraß ℘-function, which is the expression  ω ℘( ). λ ω λ ∈P \Ω

In the previous line, P stands for a period parallelepiped of the associated translation group. The complex division values of the Weierstraß ℘-function and its trace are closely related to important fundamental problems   from algebraic number theory. Putting g2 = 60 ω∈Ω\{0} ω −4 and g3 = 140 ω∈Ω\{0} ω −6 , then all complex division values of the associated normalized doubly periodic Weierstraß √ 2 g3 −D] and function P(z) := g3g−27g 2 ℘(z) lie in abelian Galois field extensions of Q[ 2

3

84

Chapter 2. Clifford-analytic Eisenstein series for translation groups

√ its trace is an element from Q[ −D]. This property provides an analogy to the number theoretical behavior of that of the division values of the exponential function which lie in abelian Galois field extensions of Q. For this reason, the classical elliptic functions play a fundamental role in class field theory. This is illuminated in detail in some of R. Fueter’s early works, see for instance [52], [53]. To give also a reference to one of the modern textbooks dealing with this problem, we refer for instance to [149] among others. Hilbert’s famous twelfth problem from [75] which deals with the problem of the construction of abelian Galois field extensions to an arbitrarily given algebraic number field provided a relatively strong motivation to ask for higher dimensional analogies of the concept of complex multiplication of the elliptic functions. In the first decades of the twentieth century, E. Hecke dedicated himself to work on generalizations of the concept of complex multiplication to the setting of several complex variables using Abelian functions (see e.g. [70, 71]). However, in the context of several complex variables theory, the Riemann condition imposes an additional restriction to the choice of the periods of the Abelian functions. The Riemann condition is a quadratic relation that the generators of the period lattices have to satisfy. It causes important restrictions for the computations of the class fields (see [70, 71] and also [63]). R. Fueter outlined at the end of his mathematical career in [62, 63] an ansatz for a quaternionic multiplication within the context of the monogenic four-fold periodic quaternionic generalizations of the Weierstraß ℘-function and the monogenic quasi-periodic quaternionic Weierstraß ζ-function. If one chooses the four primitive quaternionic periods for the period lattice Ω from a maximal integral domain that lies in a quaternionic Brandt algebra (cf. [14, 57]), then all λ, µ ∈ Ω satisfy λΩ ⊆ Ω, Ωµ ⊆ Ω, and hence also λΩµ ⊆ Ω. In view of the non-commutativity of the quaternions it is indeed strongly motivated to consider multipliers from the left and from the right. This approach involves in particular lattices where the four primitive periods are quaternions whose real components are elements from biquadratic number fields. If f (z) is a four-fold periodic monogenic function that is associated to such a particular quaternionic lattice Ω, then the function fλ,µ (z) := µf (λzµ)λ is again a four-fold periodic monogenic function associated to the same Ω for all λ, µ ∈ Ω. According to Theorem 2.5 every four-fold periodic 1-monogenic elliptic function with non-essential singularities can be represented on the other hand in terms of a (4) finite sum of the elementary four-fold periodic 1-monogenic function series
0,1;a,b (4)

and
m,1 with |m| ≥ 1. If Ω is a lattice with quaternionic multiplication, then fλ,µ can in turn be represented in terms of the same elementary generalized Weierstraß functions (associated to the same lattice) as f . In analogy to the complex case, this leads to important functional equations. In particular, one obtains explicit multiplication formulas for the quaternionic monogenic versions of the Weierstraß ℘-function — involving quaternionic multipliers. These formulas in turn lead to closed formulas

2.7. Lattices with hypercomplex multiplication

85

for the trace of the division values of the quaternionic versions of the ℘-function (4) (i.e., ℘τ (i),1 =
τ (i),1 ) from which R. Fueter drew the conclusion that the sum of  the quaternionic division values λ−1 ωµ−1 ∈P \Ω ℘τ (i),1 (λ−1 ωµ−1 ) is a quaternion whose real components all lie in a field that is generated by the real components of the involved primitive periods ω1 , ω2 , ω3 , ω4 (which stem in the simplest nontrivial case from a biquadratic number field) and by the real components of the quasi-periodicity constants ηh of the generalized quaternionic monogenic Weierstraß ζ-function ζ(z, Ω) which are ηh := 2ζ(ωh /2, Ω), h = 1, 2, 3, 4. A deeper and more concrete analysis has not been provided by R. Fueter. Nevertheless, his ideas might be rather promising: While the periods of the Abelian functions have to satisfy the Riemann condition, the periods of the quaternionic monogenic elliptic functions can be chosen absolutely free within the Brandt algebra. Every arbitrary ideal from an integral domain of a Brandt algebra provides us with four periods. One only has to take care in the sense of choosing four IR-linear independent periods ω1 , . . . , ω4 . It shall be noticed that the Abelian functions of two complex variables can be reconstructed from particular candidates of four-fold periodic monogenic elliptic functions, as mentioned for instance in [64]. Unfortunately the idea to link special monogenic functions with problems from class field theory or with general problems of number theoretical origin has been neglected after R. Fueter’s death. In this section we present extensions of R. Fueter’s ideas from [62] and [63] to the setting of the arbitrary finite dimensional Euclidean spaces Ak+1 and to more general function classes. From the very beginning it is not obvious at all whether complex and quaternionic multiplication can be generalized to arbitrary higher dimensional settings within the context of Clifford-analytic multiperiodic functions which are defined in Ak+1 . Note that both C and IH are closed under multiplication. However, if λ, ω, µ are more generally paravectors from Ak+1 , then neither products of the form λω, nor ωµ, nor λωµ remain in Ak+1 in general. It was possible to define Zorders in rational Clifford algebras Cl0k for general k (see e.g., [44, 45]); however, their restriction to the paravector space Ak+1 is not closed under multiplication within Ak+1 . It is thus not immediately so evident how the concept of complex and quaternionic multiplication of the Weierstraß functions can be adapted to the setting of the Clifford-analytic Weierstraß functions defined in the paravector space Ak+1 . In our recent paper [92] we illustrated a meaningful way to do so. First we started with the following definition: Definition 2.21 (Lattices with paravector multiplication). Let 1 ≤ p ≤ k + 1 and Ωp = Zω1 + · · · + Zωp be a p-dimensional lattice in Ak+1 . Then we say that Ωp has paravector multiplication if there are paravectors λ, µ ∈ Ak+1 where at least λ or at least µ is an element from Ak+1 \Z such that λΩp µ ⊂ Ωp .

(2.73)

86

Chapter 2. Clifford-analytic Eisenstein series for translation groups

The simplest examples of lattices in Ak+1 that satisfy this condition are rectangular lattices: e1 (Zα0 +Zα1 e1 +· · ·+Zαk ek )e1 = Zα0 +Zα1 e1 +· · ·+Zαk ek ; α0 , . . . , αk ∈ IR\{0}. (2.74) However, a special focus shall be put on those lattices that satisfy (2.73) in particular for multipliers λ, µ that are elements from the proper lattice. The following theorem provides us with a number of non-trivial examples of lattices in Ak+1 that have a paravector multiplication where the multipliers stem from the proper lattice. Theorem 2.22. Let p be an arbitrary but fixed integer with 0 ≤ p ≤ k. Let further Ωp+1 := Z + Zω1 + · · · + Zωp be a lattice in Ak+1 where the periods ω1 , . . . , ωp satisfy all N (ωi ) ∈ Z,

S(ωi ) ∈ Z,

2ωi , ωj ∈ Z

∀i, j ∈ {1, . . . , p}.

(2.75)

Then Ωp+1 has paravector multiplication. In particular, it is stable under the conjugation anti-automorphism (i.e., Ωp+1 = Ωp+1 ) and each η ∈ Ωp+1 and α ∈ Z satisfies (2.76) α ηΩp+1 η ⊂ Ωp+1 . Conversely, if Ωp+1 = Z + Zω1 + · · · + Zωp is a (p + 1)-dimensional lattice that has the property that αηΩp+1 η ⊆ Ωp+1 is satisfied for any α ∈ Z and any arbitrary η ∈ Ωp+1 , then the primitive periods satisfy all (2.75). p Proof. Let ω ∈ Ωp+1 . Then there are α0 , . . . , αp ∈ Z so that ω = α0 + j=1 αj ωj . p Furthermore, ω = S(ω) − ω. Since S(ω) = 2α0 + j=1 αj S(ωj ) ∈ Z, we can conclude directly that ω ∈ Ωp+1 . The lattice Ωp+1 is indeed stable under the conjugation anti-automorphism. Next let us prove that every η ∈ Ωp+1 satisfies ηωη ∈ Ωp+1 for any arbitrary ω ∈ Ωp+1 from which (2.76) then readily follows. One obtains ηωη = S(ηω)η − N (η)ω = 2η, ω η − N (η)ω.

(2.77)

Observe that both η, ω ∈ Ωp+1 . Therefore, there are integers α0 , . . . , αp , β0 , . . ., βp ∈ Z with p p     2η, ω = 2 α0 + αj ωj , β0 + βj ωj = j=1

j=1

 0≤i,j≤p

αi βj · 2ωi , ωj

  ∈Z

where ω0 := e0 and ω1 , . . . , ωp are the primitive periods of Ωp+1 . In view of (2.75), one has S(ηω) ∈ Z for any ω, η ∈ Ωp+1 . Putting ω = η in the above calculation yields N (η) = η, η ∈ Z. Consequently, we can write ηωη = γη − δω with integers γ, δ ∈ Z. Hence, ηωη ∈ Ωp+1 .

2.7. Lattices with hypercomplex multiplication

87

Let us now suppose that we are dealing with a lattice Ωp+1 := Z + Zω1 + · · · + Zωp that satisfies αηΩp+1 η ⊂ Ωp+1 for all α ∈ Z and for all η ∈ Ωp+1 . A lattice with these properties satisfies then in particular ωi ωj ωi ∈ Ωp+1 for the primitive periods. The relation (2.77) implies that ωi ωj ωi = (S(ωi )S(ωj ) − 2ωi , ωj )ωi − N (ωi )ωj . The product ωi ωj ωi has thus the form aωi + bωj . From ωi ωj ωi ∈ Ωp+1 follows that both elements a and b have to be integers. Hence N (ωi ) ∈ Z for all i = 0, 1, . . . , p. Further, (2.78) S(ωi )S(ωj ) − 2ωi , ωj ∈ Z for all i, j ∈ {0, . . . , p}. Inserting in particular ωj = 1 into (2.78), implies conse
quently that S(ωi ) ∈ Z. Hence, 2ωi , ωj must also be an integer. Notice once more that the multipliers do not form anymore a ring within Ak+1 for general k. In general, the multiplication of the multipliers is only closed within the whole Clifford algebra over Ak+1 , where we have zero-divisors. Nevertheless, we have the impression that the lack of the closure of multiplication within Ak+1 does not really lead to a serious obstacle. Next we give a number of important examples of paravector lattices that have the properties from Theorem 2.22: Proposition 2.23. Let Ωp+1 be a p + 1-dimensional lattice in Ak+1 of the special form Ωp+1 = Z + Zω1 + · · · + Zωp where the primitive periods ω1 , . . . , ωp have the form ωi =

k 

√ αij mj ej

(2.79)

j=0

where αij are all integers, m0 = 1 and m1 , . . . , mk are positive integers which may be square-free. Then Ωp+1 has paravector multiplication and for all α ∈ Z and all η ∈ Ωp+1 we have αηΩp+1 η ⊆ Ωp+1 . Proof. Consider simply N (ω0 ) = N (e0 ) = 1 and for i = 1, . . . , p: N (ωi ) =

k 

 √ (αij mj )2 = αi2j mj ∈ Z. k

j=0

j=0

Furthermore, S(ω0 ) = 2, and for i = 1, . . . , p we have S(ωi ) = 2αi0 ∈ Z. Obviously, ω0 , ωj = ωj , ω0 ∈ Z for all j = 0, . . . , p. For general i, l ∈ {1, . . . , p} we have 2ωi , ωl =

k  j=0

αij αlj mj ∈ Z.




88

Chapter 2. Clifford-analytic Eisenstein series for translation groups

Notice that the condition αij ∈ Z is only a sufficient condition. Indeed, it is not difficult to work out examples where the parameters αij stem from Q\Z, as given next in (2.82). In 2n -dimensional Euclidean spaces the following types of lattices play a particular role within the theory of paravector multiplication. In the four-dimensional case, i.e., n = 2, consider lattices with periods of the form (see also [63]) √ √ √ √ ωi = αi0 + αi1 m1 e1 + αi2 m2 e2 + αi3 m1 m2 e3

(2.80)

where m1 and m2 are two distinct positive square-free integers and where the elements αij are chosen in a way that (2.75) is satisfied. In the particular case where all the parameters αij are integers, the conditions (2.75) are always satisfied — independently from the choice of the parameters αij . In the eight-dimensional case consider lattices with primitive generators of the form ωi

√ √ √ = αi0 + αi1 m1 e1 + αi2 m2 e2 + αi3 m3 e3 √ √ √ √ √ √ +αi4 m1 m2 e4 + αi5 m1 m3 e5 + αi6 m2 m3 e6 √ √ √ +αi7 m1 m2 m3 e7

(2.81)

and similar lattices of dimensions 2n with n > 3 — claiming the analogous conditions on the elements m1 , m2 , . . . and αij . If one treats the case n = 2 with quaternions, i.e., by identifying the elements ei with the quaternionic imaginary units, e1 = i, e2 = j, e3 = k = ij, then the primitive periods of the quaternionic lattice (2.80) generate an ideal within an integral quaternionic Brandt algebra, as R. Fueter mentioned in [63]. The multiplication of two primitive periods from a quaternionic lattice of the form (2.80) produces again a quaternion that has the form √ √ √ √ βi0 + βi1 m1 e1 + βi2 m2 e2 + βi3 m1 m2 e3 with parameters βij from Q. If in particular all αij ∈ Z, then all the elements βij are again integers. If the elements αij from (2.80) are properly chosen so that the ideal generated by the primitive periods is furthermore a maximal integral domain in the underlying quaternionic Brandt algebra, then the associated lattice is closed under multiplication. Then λΩ ⊆ Ω, Ωµ ⊆ Ω, and thus λΩµ ⊆ Ω for all elements λ, µ ∈ Ω. Choosing two non-real linearly independent primitive periods ω1 and ω2 of the form (2.80) that satisfy moreover ω1 ω2 = ω2 ω1 , then, following [57], [1, ω1 , ω2 , ω1 ω2 ] is a basis for a maximal integral domain whenever N (ω1 ω2 − ω2 ω1 ) = 4| det W |.

2.7. Lattices with hypercomplex multiplication

89

Here W stands for the period matrix of the primitive periods (1, ω1 , ω2 , ω1 ω2 ). In this case the associated lattice Ω = Z + Zω1 + Zω2 + Zω1 ω2 is closed under multiplication. To give one explicit example for a non-rectangular quaternionic lattice that is closed under multiplication, take for instance as generating periods ω0 = 1,

ω3 = − 21 2 +

√

3 2 e1 , √ √ 7 3 5 2 e1 + 2 e2

ω1 =

1 2

+

√ √ ω2 = 7 3e1 + 5e2 , √

+

15 2 e3

(2.82)

= ω1 ω2 .

By a direct computation one can readily show that the conditions (2.75) are all satisfied, that ω1 ω2 − ω2 ω1 = 0 and that additionally N (ω1 ω2 − ω2 ω1 ) = 4| det W |. These periods generate actually a maximal integral domain within an integral quaternionic Brandt algebra. The associated lattice is hence closed under multiplication. In the eight-dimensional case, n = 3, one can embed the lattice (2.81) into the octonions O. In this sense simply identify the elements e1 , . . . , e7 with the octononic imaginary units, i.e. e1 = i, e2 = j, e3 = w, e4 = k, e5 = iw, e6 = jw, e7 = kw. Here i, j, k are the quaternionic imaginary units, while w is a further independent octonionic imaginary unit that satisfies w2 = −1, as introduced in Chapter 1.1. The multiplication of two primitive periods from an octonionic lattice that has the form (2.81) results again in an octonion of the form √ √ √ √ √ γi0 + γi1 m1 e1 + γi2 m2 e2 + γi3 m3 e3 + γi4 m1 m2 e4 √ √ √ √ √ √ √ +γi5 m1 m3 e5 + γi6 m2 m3 e6 + γi7 m1 m2 m3 e7 , where the coefficients γij are all elements from Q. In the particular case where all the elements αij are integers, the coefficients γij are integers, too. In analogy to the quaternionic case, the primitive periods generate an ideal of integral elements within the octonions. In this context, the notions “ideal” and “integral domain” shall be understood in a wider sense, having no longer the associativity. For more details about the general theory of ideals in octonions, see for instance [28]. If we choose the elements αij from (2.81) properly so that the ideal that is generated by the primitive periods is moreover a maximal integral domain, then we have in analogy to the quaternionic case that λΩ ⊆ Ω, Ωµ ⊆ Ω, and hence (λΩ)µ ⊆ Ω and λ(Ωµ) ⊆ Ω for all elements λ, µ ∈ Ω. Such a relation does in general not exist in the context of working in the eight-dimensional paravector space A8 . At this point we wish to point out once more that the octonions do not form a Clifford algebra, since they are non-associative. However, as mentioned in Chapter 1.1, they still form a normed division algebra. Higher dimensional Cayley–Dickson algebras where n > 3 do not form normed algebras anymore. Therefore, the notion of Brandt algebras cannot be transferred so directly to the setting of higher dimensional Cayley–Dickson algebras that extend the octonions

90

Chapter 2. Clifford-analytic Eisenstein series for translation groups

without imposing further restrictions. The special cases where the dimension of the underlying Euclidean space is either 2, 4 or 8 should therefore be regarded as very special subcases within the theory of hypercomplex multiplication. Now let us assume that Ω is an arbitrary lattice with hypercomplex multiplication of the form that λΩµ ⊂ Ω for some paravectors from Ak+1 , quaternions or octonions respectively. Under this condition we have the following situation: If f satisfies f (z + ω) = f (z) for all ω ∈ Ωp , then the function fλ,µ (z) := µf (λzµ)λ satisfies also fλ,µ (z + ω) = fλ,µ (z) for all ω from the same lattice. If f : Ak+1 → Cl0k is an arbitrary left and right s-monogenic function and if λ is an arbitrary paravector from Ak+1 , then in general the expression fλ,µ (z) remains only left and right s-monogenic, if µ = αλ with a real α ∈ IR. This provides a strong function theoretic motivation to regard exactly those lattices that are described in Theorem 2.22 as the canonical ones within the theory of paravector multiplication of the Clifford analytic multiperiodic functions defined in Ak+1 for arbitrary dimensions. We observe here an important connection point between Clifford analysis and number theory. Notice once more the particular role of the quaternionic and octonionic case: If f is left and right quaternionic (octonionic) s-monogenic then fλ,µ (z) is also left and right s-monogenic, for all quaternions (octonions) λ and µ. This special property is a consequence of the fact that both the quaternions and the octonions do still form normed algebras. Let us now describe in detail the paravector multiplication of the monogenic elliptic functions in arbitrary dimensions in the Clifford analysis setting. What follows provides an extension of R. Fueter’s result for the quaternionic case. The octonionic case is also treated implicitly within the following calculations. In this (p) sense, notice that the Eisenstein series
m,s can be introduced directly into the octonionic setting. One simply has to substitute the elements e0 , e1 , . . . , e7 by the eight units of the octonions. The series that we subsequently obtain — having l f = 0 formally the same form as their Clifford counterparts — satisfy then DO where DO means the octonionic Cauchy–Riemann operator as used in [68, 114, 109, 110] and elsewhere. In view of the relation (ab)b = b(ba) = a b 2 = a(bb) = a(bb) for all a, b ∈ O, as a consequence of Theorem 4.1 from [164], all the following calculations can be directly adapted to the octonionic case putting the brackets in accordance to the octonionic lattice multiplications that are considered, either of the form (λΩ)µ or λ(Ωµ). To proceed, recall that every Ωk+1 -periodic monogenic function f that has at most unessential singularities can be represented in terms of a finite sum of the (k+1) associated to this lattice. monogenic k + 1-fold periodic Eisenstein series
m This is the statement of Theorem 2.5. If Ωk+1 is a lattice that satisfies (2.75), then

2.7. Lattices with hypercomplex multiplication

91

fαλ,λ can consequently be represented in terms of a finite sum of the same se(k+1) ries
m associated with this lattice. This leads to explicit relationships between (k+1) (k+1) αλ
m (αλzλ)λ and
m (z) for arbitrary λ ∈ Ωk+1 and α ∈ Z. In particular we obtain an interesting generalization of the multiplication formula for the generalized ℘-functions from Theorem 2.18 for paravector multipliers. The following argumentation is more complicated than in Section 2.5 due (s) to the fact that the functions qm are only IR-homogeneous and not Ak+1 -homogeneous functions. Now consider the generalized monogenic Weierstraß ζ-function ζ(z, Ωk+1 ) := ζ0,1 (z, Ωk+1 ) where we assume that Ωk+1 is a lattice whose primitive periods satisfy the conditions (2.75), as for example, lattices of the form (2.79), (2.80), (2.81). Next let us take an arbitrary non-zero λ of such a lattice and an arbitrary non-zero positive integer α. Then αλzλ ∈ Ak+1 \Ωk+1 for every z ∈ Ak+1 \Ωk+1 and αλωλ ∈ Ωk+1 for every ω ∈ Ωk+1 . Hence ζαλ,λ (z) := αλζ(αλzλ, Ωk+1 )λ is a well-defined quasi (k + 1)-fold periodic function with respect to Ωk+1 . It is left and right monogenic in Ak+1 \ α1 λ−1 Ωk+1 λ−1 . Notice that the proper expression ζ(αλzλ, Ωk+1 ) is not monogenic anymore whenever λ ∈ IR. However, the factor λ on the left-hand side of this expression provides the left monogenicity, and the other one standing on the right-hand side, provides the right monogenicity. Notice that ζαλ,λ (z) has only point singularities of order k at the points 1 −1 −1 λ ωλ , ω ∈ Ωk+1 . α √ Each period parallelepiped contains {µ} = N ( αλ)k+1 -many point singularities (all of order k). The set of singularities that are contained in the fundamental period parallelepiped (including the origin) will be denoted by V in all that follows. Next we expand the function ζαλ,λ into a Laurent series in a sufficiently small neighborhood around zero: µ :=

αλζ(αλzλ)λ = αλq0 (αλzλ)λ + φ(z) = αλ = q0 (z) 

αλzλ λ + φ(z)

αλzλ k+1

1

k−1 + φ(z). αN (λ)

Here φ stands for a function that is monogenic in a neighborhood of the origin. On the other hand,  ζ(z + µ) = q0 (z) + Ψ(z) µ∈V

in a sufficiently small neighborhood around the origin, where Ψ stands analogously for a function that is monogenic in that neighborhood. The function  1 f (z) := αλζ(αλzλ, Ωk+1 )λ − ζ(z + µ, Ωk+1 ) (2.83) k−1 [αN (λ)] µ∈V

92

Chapter 2. Clifford-analytic Eisenstein series for translation groups

is thus left and right monogenic at the origin. By analogous arguments one concludes that f is also right and left monogenic at each µ ∈ V. Let i ∈ {1, . . . , k} and let us for simplicity denote the partial derivatives of f with respect to the variable xi by fτ (i) . From the quasi-periodicity property of the generalized Weierstraß ζ-function with respect to Ωk+1 , the k functions  ∂ 1 fτ (i) (z) := αλ[ ζ(αλzλ, Ωk+1 )]λ − ℘τ (i) (z + µ, Ωk+1 ) (2.84) ∂xi [αN (λ)]k−1 µ∈V

turn out to be all (k + 1)-fold periodic with respect to Ωk+1 . Since f has no singularities in each period parallelepiped, the function fτ (i) has no singularities in each period cell, either. Thus, fτ (i) (z) ≡ Ei is constant for all i = 1, . . . , k. In view of the monogenicity of ζ we hence obtain: αλζ(αλzλ)λ =

k   1 ζ(z + µ) + Ej Vτ (j) (z) + E. [αN (λ)]k−1 j=1

(2.85)

µ∈V

Applying again a shift of the form (z → z + ωh ), h = 0, . . . , k on the argument of both sides of the previous equation yields αλζ(αλzλ + αλωh λ)λ =

k   1 ζ(z + µ + ω ) + Ej Vτ (j) (z + ωh ) + E h [αN (λ)]k−1 j=1 µ∈V

(2.86) for all h = 0, 1, . . . , k. The special property that Ωk+1 is a lattice with hypercomplex multiplication allows us to conclude that there are integers nhj ∈ Z with αλωh λ =

k 

nhj ωj .

(2.87)

j=0

Notice here, that we do not obtain such a relation for general lattices. Applying (2.21) successively yields ζ(αλzλ + αλωh λ) = ζ(αλzλ) +

k 

nhj ηj .

j=0 [1]

For simplicity we write here ηj for ηj . Next combine this relation with (2.86). After this one gets in view of (2.85): αλ

k



 nhj ηj λ =

j=0

k

   1 + η Ej Vτ (j) (ωh ). h [αN (λ)]k−1 j=1 µ∈V

√ From V = N ( αλ)k+1 follows that the previous equation simplifies to αλ

k

 j=0

k   √ nhj ηj λ − N ( αλ)2 ηh = Ej (ωhj − ej ωh0 ) j=1

(2.88)

2.7. Lattices with hypercomplex multiplication

93

k where we write again ωh = j=0 ωhj ej . Let us use the notation det W for the determinant of the period matrix W , as defined in (2.65), and Θhj for the adjoint determinant associated with the element ωhj . Since (2.88) is satisfied, the following equation is also satisfied: k  k k  k k  

    √ Θhi αλ nhj ηj λ − N ( αλ)2 Θhi ηh = Ej (Θhi ωhj − ej Θhi ωh0 ). j=0

h=0

h=0

h=0 j=1

(2.89) Applying (2.66) to (2.89), yields for all i = 1, . . . , k:  

 k  k k   √ Θhi αλ nhj ηj λ − N ( αλ)2 Θhi ηh Ei =

h=0

j=0

h=0

det W

.

(2.90)

In view of the linear independence of the primitive periods, we have det W = 0, so that we actually may divide by det W . Notice further that the elements ηj cannot all vanish, otherwise the ζ-function would be (k + 1)-fold periodic. This, however, would be a contradiction to Theorem 2.8 from Section 2.3. Hence the values Ej need not all vanish. In the context considered in this section, we get thus a difference to the integer multiplication formulas that we described in Section 2.5. Taking the limit z → 0 on both sides of equation (2.84) provides us with an explicit formula for the trace of hypercomplex division values of the generalized monogenic ℘-functions: Theorem 2.24. Let i ∈ {1, . . . , k}. Let Ωk+1 be a lattice whose primitive periods satisfy (2.75). Further let α ∈ IN, λ ∈ Ωk+1 \{0} and V = {µ ∈ F | µ = 1 −1 ωλ−1 , ω ∈ Ωk+1 } where F stands for the fundamental period parallelepiped. αλ Then 

℘τ (j) (µ) =

µ∈V\{0}

k k k

     √ [αN (λ)]k−1   nhj ηj λ − N ( αλ)2 Θhi ηh . Θhi αλ det W j=0 h=0

h=0

(2.91) The trace of the hypercomplex division points of the generalized monogenic ℘functions is a paravector from Ak+1 whose real components all lie in the field that is generated by the number field of the real components of the primitive periods ωh and by the real components of the quasi-periodicity constants ηh of the monogenic Weierstraß ζ-function. If we have a lattice of the form (2.79) with m1 , . . . , mk being all mutually distinct and square-free, then the real components of the primitive periods ωh √ √ generate the field Q[ m1 , . . . , mk ]. The quaternionic or octonionic components of the periods of the special lattices (2.80) or (2.81), respectively, generate a biquadratic or triquadratic number field, respectively. To obtain an exhaustive number theoretical description of the trace of the hypercomplex division values of the

94

Chapter 2. Clifford-analytic Eisenstein series for translation groups

generalized ℘-functions, a deeper number theoretical analysis of the constants ηh will be required. Notice further, that we obtain by applying on both sides of (2.86) differentiation arguments that αλ

 (k+1) 1 ∂ |m|+|r| ζ(αλzλ, Ωk+1 )λ −
m+r (z + µ, Ωk+1 ) = 0. m+r k−1 ∂x [αN (λ)] µ∈V

A similar limit argument as presented previously provides us with formulas for the (k+1) trace of the division values of the series
m . In particular, we obtain 
(k+1) (µ) = 0 m µ∈V\{0}

for all multi-indices m where |m| is an even positive integer. Next let us treat the harmonic case. In this context keep in mind that the  (2) series ω∈Ωk+1 qm (z + ω) only converges normally up from |m| ≥ 3 (instead for |m| ≥ 2 for s = 1). The harmonic analogues of the Weierstraß ℘-function in (k+1) this setting are given by the series
m,2 where m is a multi-index of length 2. (2)

(1)

From Dqτ (i) (z) = (1 − k)qτ (i) (z) follows that for each m = τ (i) + τ (j) with i, j ∈ {0, 1, . . . , k}, k 

(k+1)

(k+1)


τ (i)+τ (j),2 (z)ej = (1 − k)
τ (i),1 (z).

j=0 (k+1)

The functions
τ (i)+τ (j),2 which are scalar-valued coincide thus up to the real constant ±(1 − k) with the real components of the monogenic paravector-valued (k+1) series
τ (i),1 (z). This admits an immediate transfer of the results obtained in the monogenic case to the harmonic case. This direct argument can only partially be adapted to the setting D[s] f = 0 where s > 2. With the formula D

[β] (s) qm (z)

(s−β) = C(k)qm (z),

where C(k) stands for a real constant that depends only on k, one can reduce the analysis of a number of s-monogenic variants of associated Weierstraß ℘-functions to the study of the monogenic ones. However, the relations become more complicated the larger s becomes. This relation cannot be used directly to draw conclusions for arbitrary choices of m, as for instance for the index 3τ (i) (i = 1, . . . , k) in (s) the setting s = 3, as one may easily verify. The inhomogeneity of qm with respect to Ak+1 for all |m| ≥ 1 is a significant obstacle for a direct derivation of multiplication formulas for all configurations of s and m. To discuss a further interesting

2.7. Lattices with hypercomplex multiplication

95

positive example consider in Ak+1 with k ≡ 1 mod 2 the special holomorphic Cliffordian ℘-functions (cf. [105]) (k+1)

[k]


(k−1,0,...,0)+τ (i) (z) =: ℘τ (i) (z)

i = 0, . . . , k

where (k − 1, 0, . . . , 0) is a multi-index whose first entry refers to the differentiation with respect to the x0 -direction. These functions stem from one global holomorphic Cliffordian primitive, which is here for simplicity denoted by ζ [k] (z) = ζ(k−1,0,...,0),k (z). We have ∂ [k] [k] ζ (z) = ℘τ (i) (z). ∂xi

(2.92)

Consequently, there are paravector constants βh with ζ [k] (z + ωh ) = ζ [k] (z) + βh for h = 0, . . . , k. If we deal with a lattice that satisfies the conditions from (2.75), then we can proceed in a similar way as in the proof for Theorem 2.24 in order to establish an explicit formula for the trace of their hypercomplex division points, however, involving one further restriction: [k]

Theorem 2.25. Let i ∈ {0, 1, . . . , k}, k be an odd positive integer, and ℘τ (i) (z) be an arbitrary function of those from (2.92). Let Ωk+1 be a lattice whose primitive periods satisfy (2.75). For all λ ∈ Ωk+1 with Sc(λ) = 0 we have  [k] ℘τ (i) (µ) µ∈V\{0}



=

k−1 k 

 − N (λ) det W

Θhi λ

k



k     nhj βj λ − (−1)k−1 N (λ)2 Θhi βh .

j=0

h=0

h=0

(2.93) Proof. Notice first that the Laurent expansion of the particular function ζ [k] (z) has the special form ζ [k] (z)

= =

∂ k−1 (k) q0 (z) + Φ(z) ∂xk−1 0

(−1)k−1 (k − 1)!z −k + Φ(z).

Under the condition that Sc(λ) = 0 we have the special relation (λzλ)−k = (−1)k−1 λ−1 z −k λ−1 Hence, λζ [k] (λzλ)λ = (k − 1)!

1 . N (λ)k−1

z −k + λΦ(λzλ)λ, N (λ)k−1

96

Chapter 2. Clifford-analytic Eisenstein series for translation groups

where the function Ψ(z) := λΦ(λzλ)λ is left and right holomorphic Cliffordian in an open neighborhood around zero. The function f (z) = λζ [k] (λzλ)λ −

(−1)k−1  [k] ζ (z + µ) N (λ)k−1

(2.94)

µ∈V

is left and right holomorphic Cliffordian at zero and also at the other points µ ∈ V. Its k + 1 partial derivatives fτ (i) (z) = λ[

∂ (−1)k−1  [k] {ζ [k] (λzλ)}]λ − ℘τ (i) (z + µ) i = 0, . . . , k, (2.95) ∂xi N (λ)k−1 µ∈V

are then (k + 1)-fold periodic holomorphic Cliffordian functions in the whole space Ak+1 . From the first Liouville theorem for holomorphic Cliffordian functions (cf. [105]) follows that fτ (i) (z) ≡ Ei and hence λζ [k] (λzλ)λ =

(−1)k−1  [k] ζ (z + µ) + E0 x0 + E1 x1 + · · · + Ek xk + E, N (λ)k−1 µ∈V

involving a further independent Clifford constant E0 , since f is not monogenic. Similar arguments as in the proof of Theorem 2.24 can now be applied so that one finally arrives at the stated result.
Remarks. Qualitatively, the formula (2.93) has from the formal point of view a similar structure as (2.91). However, notice that we put an additional restriction on the multipliers λ, namely that Sc(λ) = 0. Note in this context that the particular lattices (2.79), (2.80) and (2.81) contain infinitely many lattice points with no scalar part. In general the elements βh do not coincide with the quasi-periodicity constants ηh from the monogenic ζ-function. Applying to all these cases a differentiation argument provides us consequently with formulas for the sum of the hypercomplex division values of the partial derivatives of these functions. (s)

As a consequence of the inhomogeneity of qm (z) with respect to Ak+1 for |m| ≥ 1, this argument cannot be adapted that directly to the functions (k+1) (k+1)
(0,k−1,...,0)+τ (i) (z),
(0,0,k−1,0,...,0)+τ (i) (z), etc.. In those cases the Laurent expansion of the associated primitive ζ˜[k] reads ζ˜[k] (z) = (−1)k−1 z −1 (ej z −1 )k−1 + Φ(z) and hence λζ˜[k] (λzλ)λ = (−1)k−1 λ(λ−1 z −1 λ−1 )(ej λ−1 z −1 λ−1 )k−1 λ + λΦ(λzλ)λ. Whenever j = 0 it is not possible simply to shift all the λ terms to the left or to the right-hand side of terms which only contain powers of z.

2.8. Bergman kernels of rectangular domains

97

We conclude this section by dedicating a few summarizing words to the hypercomplex multiplication of the translative Eisenstein series associated to a period lattice of dimension p < k + 1. If p < k + 1 − s, then the function series  (s) qm (z + ω) ω∈Ωp

is already convergent up from m = 0. The same holds for the expression    (s) (s) (s) q0 (z) + q0 (z + ω) − q0 (ω) ω∈Ωp \{0}

in the case p = k +1−s, as mentioned previously. In all these cases one can exploit (s) the particular Ak+1 -homogeneity of the functions q0 which we do not have for the cases with |m| ≥ 1, as mentioned a few lines above.  In the 1-monogenic case, in which the series ω∈Ωp q0 (z + ω) is convergent for all p ≤ k − 1, this allows us readily to establish (p)

λ
0 (λzλ)λ

=

 ω∈Ωp

=



µ∈V

1 q0 (z + λ−1 ωλ−1 ) N (λ)k−1

1 (p)
(z + µ) N (λ)k−1 0

for all λ ∈ Ωp .

(2.96)

A similar formula will be obtained for the case s = 2 with p ≤ k − 2, for the case s = 3 with p ≤ k − 3, etc. In the case s = 1 and p = k one can establish an analogous formula, using a generalized version of Lemma 2.16 for the multi-index m = 0. The cases s = 2n + 1, p = k − 2n can be treated in a similar way, since (s) q0 is odd whenever s is odd. In the cases where s = 2n, p = k − 2n + 1 it is not (s) allowed to apply Lemma 2.16, since in these functions q0 are even. However, one may immediately infer a similar multiplication formula up to a constant. These formulas give rise to explicit formulas for the sum of the hypercomplex division (p) values of the associated series
m,s . Differentiation arguments and an analogous limit argument lead further to formulas for the trace of the hypercomplex division (p) values of
m+r,s for |r| ≥ 1. The trace vanishes if |m| + |r| + s is odd. The remaining cases are more difficult to treat in view of the inhomogeneity (s) of the functions qm . The elegant argumentation with Liouville’s theorem cannot be applied for p < k + 1, either.

2.8

Bergman kernels of rectangular domains

In classical complex analysis, holomorphic automorphic forms are intimately related to reproducing kernel functions from the classical Bergman and Hardy spaces associated to their fundamental domains.

98

Chapter 2. Clifford-analytic Eisenstein series for translation groups

We briefly recall: The classical Bergman space B 2 (G, C) is the space of functions that are holomorphic and square-integrable over a domain G ⊂ C. The closure of the space of functions that are holomorphic over G with continuous extension to the boundary being square-integrable over the boundary ∂G is called the Hardy space and is denoted by H2 (∂G, C). Both function spaces are Hilbert spaces with a continuous point evaluation from which the existence of a uniquely defined reproducing kernel function follows. The kernel function is called the Bergman kernel in the first case and the Szeg¨o kernel in the second one. The Bergman kero kernel SG (z, w) are both functions in two complex nel BG (z, w) and the Szeg¨ variables which are holomorphic in the first variable and anti-holomorphic in the second one. They satisfy  BG (z, w)f (w)dVw f (z) = G

and

 SG (z, w)f (z)dSw

f (z) = ∂G

for any function f ∈ B 2 (G, C) or f ∈ H2 (∂G, C), respectively, where dV stands for the volume measure and dS for the Lebesgue surface measure, respectively. For details about the classical theory of these function spaces, see for instance the textbook [6]. In contrast to the behavior of the Cauchy kernel, the Bergman and the Szeg¨o kernel depend on the domain. With each domain a different Bergman and Szeg¨o kernel function is associated. The determination of explicit formulas for these reproducing kernel functions is very difficult in general. In the classical case when we deal with a simply connected domain G = C the associated kernel functions are directly related to the Riemann mapping function f , which is the function that maps G conformally onto the upper half-plane. One has BG (z, w) = −

1 f  (z)f  (w) , π (f (z) − f (w))2

(2.97)

and furthermore 2 (z, w) = SG

1 BG (z, w). 4π

(2.98)

For domains for which one knows the Riemann mapping function f explicitly, one thus obtains immediately explicit and closed formulas for the Bergman and the Szeg¨o kernel associated to G, respectively. The vertical strip S := {z = x + iy ∈ C | 0 < x < π} is mapped by the conformal map f (z) = exp(iz) onto the upper half-plane. An

2.8. Bergman kernels of rectangular domains

99

application of (2.97) leads thus immediately to BS (z, w)

= −

1 f  (z)f  (w) π (f (z) − f (w))2

exp(i(z − w)) 1 π (exp(iz) − exp(−iw))2 1 1 1 1

 = − = 2 2 π (exp(i(z + w)/2) − exp(−i(z + w)/2)) 4π sin z+w 2 = −

=

1 (1)
(z + w; 2πZ). π 2

By the same method one obtains for a rectangular domain of the form R = {z = x + iy ∈ C | 0 < x < 1, 0 < y < 1} that BR (z, w) =

 1

℘(z + w; 2Z + 2iZ) − ℘(z − w; 2Z + 2iZ) . π

The modular function j(z) =

(60G4

60G4 (z) − 27(140G6 (z))2

(z))3

maps the hyperbolic triangle F = {z = x + iy ∈ C | 0 < x
1} 2

onto the upper half-plane. Again we can conclude consequently that BF (z, w) = −

1 j  (z)j  (w) . π (j(z) − j(w))2

An application of (2.98) provides then explicit formulas for the associated Szeg¨ o kernel functions. The explicit knowledge of the Bergman and the Szeg¨o kernel for a domain gives rise to the solution of an optimization and approximation problem: The Bergman operator, defined by  [IBf ](z) = BG (z, w)f (z)dV G

where f stands now for an arbitrary L2 (G, C)-function is namely an orthogonal o projector projector from L2 (G, C) to B 2 (G, C). Similarly, the Szeg¨  SG (z, w)f (z)dS [Sf ](z) = ∂G

100

Chapter 2. Clifford-analytic Eisenstein series for translation groups

provides an orthogonal projection from L2 (∂G, C) into H2 (∂G, C). In contrast to this the Cauchy transform induces only an orthogonal projection if and only if G is the unit disk. In the 1970s R. Delanghe et al. started to study Hilbert spaces with continuous point evaluation that consist of Clifford-valued functions that are monogenic and square-integrable over a domain G ⊂ Ak+1 . These Hilbert spaces provide a canonical generalization of the classical Bergman spaces to the context of Clifford analysis. Similarly, Hardy spaces of Clifford-valued monogenic functions with boundary values in L2 (∂G, Cl0k (IR)) were introduced. See [35, 12] and the monograph [13]. The general theory of function spaces of monogenic functions being square-integrable over a domain in the Euclidean space or over its boundary respectively, has been extended by a number of authors including for example D. Constales (cf. e.g. [21, 22], M. Shapiro and N. Vasilevski (see for instance [150, 151, 163]), J. Cnops (cf. e.g., [29]), D. Calderbank [18] among many others. Meanwhile its study has become one of the main topics of Clifford analysis. Similarly, as in the complex case, one is interested in having closed and explicit formulas for the reproducing kernel functions for some special domains. Explicit formulas for the Bergman and the Szeg¨ o kernel for the unit ball can be found for instance in [13]. However, in contrast to the planar case, there is no direct higher dimensional analogue of the Riemann mapping theorem in spaces of real dimension n ≥ 3, due to the fact that the set of conformal mappings in the space is restricted to the set of M¨ obius transformations (Theorem 1.9). Only if G and G∗ are domains that can be mapped by a M¨ obius transformation onto each other, then there is an isometry between the associated Hardy spaces H2 (∂G, Cl0k (IR)) and H2 (∂G∗ , Cl0k (IR)) (cf. e.g., [21, 29, 83]). Based on the knowledge of the Szeg¨ o kernel of the unit ball, J. Cnops derived in [29] an explicit formula for the Szeg¨ o kernel of the upper halfspace of IRk . In contrast to the classical complex case there is no isometry between the corresponding Bergman spaces. In general there is no analogue of (2.98) in the higher dimensional case, either. However, within the special framework G = H + (IRk ) a particular analogue of (2.98) can be established in the form BH (x, w) = −2

∂ SH (x, w), ∂wk

(2.99)

as proved [29]. With this formula J. Cnops managed to establish a closed formula for the Bergman kernel of the upper half-space. The determination of explicit formulas for these kernel functions in the higher dimensional case is thus even more difficult than in the planar case. This explains why not many formulas are developed so far. D. Calderbank managed in 1996 to work out closed representation formulas for the kernel functions for annular domains, see [18]. J. Peetre and P. Sj¨ olin derived in [126] a non-explicit formula

2.8. Bergman kernels of rectangular domains

101

for the Szeg¨ o kernel for the special strip domain S0 = {z ∈ Ak+1 | 0 < x0 < d} in terms of plane wave integrals. In view of the complex case it is natural to ask whether perhaps the higher dimensional analogues of the translative Eisenstein series give rise to closed and explicit formulas for the kernel function associated to higher dimensional strip and rectangular domains. In the papers [25] and [24], both jointly written with D. Constales, this conjecture has been affirmed. It turned out that one can indeed express the Bergman reproducing kernel function of a rectangular domain that is (p) bounded in p-directions in terms of a finite sum of the Eisenstein series
τ (j) . In this section we summarize the main results from [25]. In the following section we will summarize the first main result from [24] where we showed that the Szeg¨ o kernel function of the strip domain S0 is explictly given by the one-fold periodic monogenic cosecant function csc1,1 (z). We start by introducing the basic setting and some special notation. Without loss of generality we consider rectangular domains in Ak+1 that are described by Rk1 ,k2 := {z ∈ Ak+1 | 0 < xj < dj , j = 0, . . . , k1 − 1, xj > 0, j = k1 , . . . , k2 − 1} where its first k1 sides (0 ≤ k1 ≤ k + 1) are assumed to be each of finite length d0 , . . . , dk1 −1 , and its sides in the following k2 − k1 dimensions (k1 ≤ k2 ≤ k + 1) are semi-infinite and its sides in the remaining directionsare infinite in both direck tions. Let K2 := {0, 1, . . . , k2 − 1}. Suppose that w = j=0 wj ej is an arbitrary paravector. Then one associates to any subset A ⊆ K2 the paravector wA whose components are defined by (wA )j = (−1)j∈A wj where (−1)j∈A = 1 if j ∈ A and (−1)j∈A = −1 if j ∈ A. Next let us abbreviate the Cauchy kernel q0 (w − z) by K(z, w) and let us use the notation K  (z, wA ) = {K(z, wA )}Dw which shall be understood in the distributional sense. Let us first treat the case k1 < k + 1. To meet our ends we first need some preparative lemmas: Lemma 2.26. For all A ⊆ K2 , (wA )Dw

w − z ( w − z 2 )Dw A

A

2

= k + 1 − 2|A|, = w − z , = 2(w − z A ),

(2.100) f or all z, w ∈ Ak+1 , f or all z, w ∈ Ak+1 .

A 2

(2.101) (2.102)

With this lemma one can show next  (z, w) satisfy Lemma 2.27. Let A ⊆ K2 . Then the distributions KA

K∅ (z, w) = δ(w − z),  KA (z, w) =

(2.103)

1 (k + 1 − 2|A|) w − z − (k + 1)(w − z)(w − z ) , (2.104) Ak+1

wA − z k+3 A

2

A

A

102

Chapter 2. Clifford-analytic Eisenstein series for translation groups

where we assume z = wA in (2.104). Furthermore  KA (z, w) = O



1

z k+1

,

z → +∞,

(2.105)

uniformly for w in a given compact set and   (z, w). KA (w, z) = KA

(2.106)

The expression K  (z, wA ) is left monogenic in the first argument z and right conjugate monogenic in w, except at z = wA . In what follows let us use the abbreviation K π (z, w) :=

1 (k1 )
(w − z) Ak+1 0

for the periodization of the Cauchy kernel with respect to the rectangular lattice 2Zd0 + · · · + 2Zdk1 −1 . The Theodorescu transform associated to K π (z, wA ) is then given by the integral  ((K π (z, wA ))Dw )f (w) dw0 · · · dwk ,

TA f (z) =

(2.107)

w∈R

where Dw is understood again in the distributional sense. Since the periodization of δ(z − w) has only one point z = w of its support belonging to R, one obtains T∅ f (z) = f (z)

(2.108)

as a consequence of (2.103). With these tools in hand one can prove the following important proposition: Proposition 2.28. Let 0 ≤ k1 < k. Let z ∈ R := Rk1 ,k2 and let U be an open neighborhood of R, the closure of R. If f : U → Cl0k (IR) is left monogenic in U , then  (−1)|A| TA f (z) = 0. (2.109) A⊆K2

Proof. From (2.107) we have  A⊆K2

(−1)|A| TA f (z) =

, 

 (−1)|A| (K π (z, wA ))Dw f (w) dw0 · · · dwk .

w∈R A⊆K2

Next one applies Stokes’ theorem in the distributional sense on this expression.

2.8. Bergman kernels of rectangular domains

103

This leads to 

(−1)|A| TA f (z) =

k 1 −1



,

j=0 w∈R,wj =dj

A⊆K2

−

k 1 −1

,

 (−1)|A| K π (z, wA ) dσw f (w)

A⊆K2



j=0 w∈R,wj =0 A⊆K2

−

k 2 −1

,



 (−1)|A| K π (z, wA ) dσw f (w)  (−1)|A| K π (z, wA ) dσw f (w)

j=k1 w∈R,wj =0 A⊆K2

, 

−

w∈R

 (−1)|A| K π (z, wA ) Df (w) dw0 · · · dwk .

  A⊆K2 =0   =0

(2.110) When wj = 0, we have trivially that wA = wA∆{j} . When wj = dj , we have wA = wA∆{j} ± 2dj . Since K π (z + 2dj ej , w) = K π (z, w) one can consequently replace K π (z, wA ) by K π (z, wA∆{j} ) in all the boundary integrals that appear in (2.110). Next, let B = A∆{j}. Summing over all A ⊆ K2 and all j = 0, . . . , k2 is equivalent to summing over all B ⊆ K2 and all j = 0, . . . , k2 , replacing A by B∆{j}. However, we have (−1)|A| = −(−1)|B| as a consequence of |A| = |B| ± 1, so that by applying Stokes’ theorem in the other direction, the right-hand side of (2.110) simplifies precisely to  − B⊆K2 (−1)|B| TB f (z),


so that the assertion follows. This proposition gives rise to the following result for the cases k1 < k + 1.

Proposition 2.29. Let k1 < k + 1. Then the Bergman kernel of R := Rk1 ,k2 has the form B(z, w)  =



(n0 ,...,nk1 −1 )∈Zk1

A⊆K2 A=∅

  (−1)|A|+1 KA (z + 2n0 d0 e0 + · · · + 2nk1 −1 dk1 −1 ek1 −1 , w) . (2.111)

Proof. The square integrability over R may be concluded by (2.104). None of the singularities z = wA , (A = ∅), lie in R, and the functions decrease fast enough to be square-integrable over the unbounded dimensions of R. The monogenicity in z follows simply by Weierstraß’ convergence theorem. Property (2.106) implies that (2.111) has the required conjugate symmetry in z and w. We are thus left to verify the reproducing property of B(z, w). In the case where f is monogenic in a neighborhood of R, the reproducing property follows  (z, w), at once from (2.109). This follows immediately from the definition of KA

104

Chapter 2. Clifford-analytic Eisenstein series for translation groups

when the term A = ∅ is separated from the others and (2.108) is used to simplify it. Let us now suppose that f is more generally an arbitrary element from L2 (R, Cl0k (IR)). Then consider the functions

 ε fε (z) = f (1 − ε)z + (d0 e0 + · · · + dk1 −1 ek1 −1 ) + ε(ek1 + · · · + ek2 −1 ) , ε > 0. 2 We observe that the function f can be approximated as closely as desired in the space L2 (R, Cl0k (IR))) by fε , ε → 0+ , which is a left monogenic function in a neighborhood of R and hence reproduced by the expression in (2.111). Taking the limit as ε → 0+ proves finally the reproducing property of the more general function f . The proposition follows from the uniqueness of the Bergman kernel.
Finally let us turn to the case k1 = k + 1. This case is more difficult to handle, since there is no meromorphic (k + 1)-fold periodic function that has only one point singularity of order k in a period cell. Therefore we cannot define a useful analogue of the auxiliary function K π in the setting k1 = k + 1. For this reason a more technical argument is required. The crucial point for the following argumentation is that (2.105) can be strengthened to

1    , z → ∞. (2.112) (−1)|A|+1 KA (z, w) + O

z k+2 A⊆K2 ,A=∅

For the detailed proof, see Lemma 2 from [25]. From (2.112) follows that also

    B(z, w) = (−1)|A|+1 KA (z + 2n0 d0 e0 + · · · + 2nk dk ek , w) (n0 ,...,nk−1 )∈Zk

A⊆K2 A=∅

(2.113) is a normally convergent series and one can prove: Proposition 2.30. In the case k1 = k + 1, B(z, w) as defined in (2.113), is the Bergman reproducing kernel of the rectangular domain R := Rk+1,0 . Proof. The qualitative idea of the proof is rather similar to that of Proposition 2.29. However, one has to take more care in the part of the proof concerning the reproducing property. To this end consider for the following partial sums of the series (2.113):

    T6N (z, w) = (−1)|A| KA (z + 2n0 d0 e0 + . . . + 2nk dk ek , w) −N 0}.

j=1

In the case p = 1, q = k − 1 consider the classical Siegel half-space H + (C1,k−1 ) = {Z = x + iy ∈ C1,k−1 | yk2 >

k−1 

yj2 , yk > 0}.

j=1

In the remaining complexified cases with p, q ≥ 2, p < q consider the domains +

p,q

H (C

) = {Z = x + iy ∈ C

p,q

|

yk2

>

p 

yj2 +

j=1

k−1 

x2j , yk > 0}

j=p+1

which of course are Siegel type domains as well. Similar domains are considered for p > q. One can verify by a direct calculation that these half-spaces have no intersection with the nullcone. As outlined in Section 1.2, M¨ obius transformations in IKp,q can be described in terms of the Vahlen groups SV (IKp,q ). The group SV (Cp,q ) contains the real Vahlen groups SV (IRr,s ) where r ≤ p, s ≤ q as subgroups. Each arbitrary discrete subgroup of SV (Cp,q ) is contained in a group that is isomorphic to ΓC (Ω2k ) where ΓC (Ω2k ) = J, Tω1 , · · · , Tω2k

146

Chapter 3. Clifford-analytic Modular Forms

where ω1 , . . . , ω2k are IR-linear independent vectors from Ck . Of course, the whole group ΓC (Ω2k ) does not act discontinuously for any arbitrary choice of lattice. Let us again restrict consideration to ω1 = e1 , . . . , ωk = ek , ωk+1 = ie1 , . . ., ω2k = iek which involves discontinuous actions. Turning first to the case IK = IR, one sees immediately that the group Γd (IRp,q ) = J, Tek−d , . . . , Tek−1 leaves H q+ (IRp,q ) invariant when d ≤ q − 1. One observes that the case where p = 0, q = k (or similarly p = k, q = 0) is special. Under this condition, H + (IRp,q ) is invariant under the modular group with the maximal number of translative generators. The situation is different in the complexified case where all the cases are equivalent. In the complexified case we always have an invariance under the inversion and a translation group generated by k linear independent translation matrices, independently from the choice of p and q. The space H q+ (C0,k ) is left invariant under Γk (C0,k ) = J, Te1 , . . . , Tek−1 , Tiek . Similarly, H q+ (C1,k−1 ) is left invariant under Γk (C1,k−1 ) = J, Te1 , . . . , Tek . Analogously, for p, q ≥ 2, the half-cone H + (Cp,q ) is invariant under Γk (Cp,q ) = J, Te1 , . . . , Tep , Tiep+1 , . . . , Tiek−1 , Tek . All these groups and subsequently also their principal congruence subgroups of any arbitrary level act discontinuously on the respective half-space. These halfspaces are then the adequate definition domains for an extension of the theory of s-monogenic modular forms related to the associated hypercomplex modular groups to the setting of arbitrary Minkowski type spaces. The conformal invariance formula from Theorem 1.28 extends in a natural way to the framework of circular half-cones and related groups considered here, simply by replacing the expression (s) q0 (cx + d) more generally by  1  s ≡ 0 mod 2, s < k = p + q,  

(cx + d)(cx + d) (k−s)/2 (s) Q0 (cx + d) = cx + d   s ≡ 1 mod 2, s < k = p + q, 

(cx + d)(cx + d) (k−s+1)/2 for x ∈ IRp,q , or by     (s) Q0 (cZ + d) =   

1 [(cZ + d)(cZ + d)](k−s)/2 cZ + d [(cZ + d)(cZ + d)](k−s+1)/2

s ≡ 0 mod 2, s < k = p + q, s ≡ 1 mod 2, s < k = p + q,

3.5. Some Perspectives

147

for Z ∈ Cp,q whenever p + q = k ≡ 0 mod 2. It is crucial to observe that the (s) (s) expressions Q0 (cx + d) and Q0 (cZ + d) are again automorphy factors. In order to construct Clp,q (IKp,q )-valued modular forms on H q+ (IKp,q ) to Γd (IKp,q )[n] that are annihilated by DIsKp,q one only has to replace in Theorem 3.4 (s) (s) (s) the expression q0 (cx + d) by Q0 (cx + d) or Q0 (cZ + d), respectively, and to ˜ consider for f adequate starting functions that are invariant under the translation group contained in Γd (IKp,q )[n], say Td (IKp,q )[n]. In the case IK = IR one obtains the convergence condition d < k − s, d ≤ q. The convergence conditions for the complexified modular forms on H q+ (Cp,q ) with respect to Γd (IKp,q )[n] are always d < k − s, independently from the signature of the space. Under these conditions, the constructions  (s) Q0 (cx + d)∗ f˜(M x ) x ∈ H + (IRp,q ) M :Γd (IRp,q )[n]\Td (IRp,q )[n]

and

 p,q

M :Γd (C

(s)

Q0 (cZ + d)∗ f˜(M Z ) p,q

)[n]\Td (C

Z ∈ H + (Cp,q )

)[n]

provide us with non-trivial examples of Clp,q (IK)-valued modular forms with respect to Γd (IRpq )[n] in Ker DIsKp,q when inserting e.g. for f˜ the function f˜ ≡ 1 (d) (d) or f˜(x) = Gm (x + β, Td (IRp,q )), x ∈ H + (IRp,q ) resp. f˜ = Gm (Z + β, Td (Cp,q )), Z ∈ H + (Cp,q ) where β may be chosen arbitrarily from H + (IKp,q ). In [89, 90] we gave elementary convergence proofs. The convergence of the complexified series in the half-cone H q+ (Cp,q ) which is the complexified extension of a series that converges in a half-space from IRk can also by proved by applying the classical extension theorems from several complex variables theory on its restriction to the proper half-space of IRk whenever the centers of the singularity cones of the complexified series lie on IRk -like domain manifolds. In this case a function series that converges on a half-space of IRk normally to a real analytic function lifts naturally to a complex-analytic function on the complexification of the half-space under the same convergence conditions. See [152, 77] for the basic theory of the general extension theorems that are needed to meet this end. Applying the same modifications, we can also obtain extensions of Theorem 3.9 and Theorem 3.11 in the context of Cartesian products of the half-spaces H q+ (IKp,q ). For the analogue of Theorem 3.9 one needs again to claim that n ≥ 3 for odd s in order to get non-trivial examples. For the detailed elementary convergence proofs and particular examples see [90] and [93].

3.5

Some Perspectives

The constructions and examples developed in Chapter 3 extend the results from Chapter 2 from the context of translation groups to that of more general arith-

148

Chapter 3. Clifford-analytic Modular Forms

metic subgroups of the Vahlen group, including generalizations of the classical modular group SL(2, Z) and their principle congruence subgroups. In turn it is realistic to expect an amplification of the range of the particular applications of the translative s-monogenic Eisenstein series to a much more general context, using s-monogenic automorphic forms for more general discrete subgroups of the Vahlen group instead. This includes the hope to arrive at closed formulas for Bergman and Szeg¨o reproducing kernel functions of a number of much more general polyhedron type domains that include rectangular and strip domains as the simplest ones. Indeed, as mentioned at the end of Chapter 3.1, the Bergman kernel function of a domain that is composed by the Cartesian product of a two-dimensional fractional wedge-shaped domain in spanIR {1, ek } and a rectangular domain from spanIR {e1 , . . . , ep } turned out actually to be a variant of an Eisenstein type series which is related to the discrete rotation group generated by the matrix diag(exp( πenk ), exp(− πenk )) and to a translation group in spanIR {e1 , . . . , ep }. This result is a first step in the suggested direction and underlines some hope that in future analogous results could be established within a more general context. While the translation invariant s-monogenic Eisenstein series from Chapter 2 provide building blocks for s-monogenic functions that are defined on conformally flat cylinders and tori in IRk , the s-monogenic automorphic forms associated to more general discrete subgroups of the Vahlen group give rise to s-monogenic functions defined on more general conformally flat spin manifolds which in turn arise from factoring out a domain from IRk by the discrete group. A central problem that arises in this context is then to look for special representatives within the classes of s-monogenic automorphic forms that give rise to Cauchy or Green kernels which admit consequently the solution of important boundary value problems on these kinds of manifolds, including in particular the Dirichlet problem. In the recent paper [97] one step has been taken in this more general direction within the framework of some particular discrete subgroups of Vahlen’s group that are different from the translation group. With special automorphic forms associated to the groups G1 = {±1} and G2 = {2k } a Cauchy kernel could be constructed for the real projective spaces P IRk ∼ = S k /G1 and for S 1 × S k−1 ∼ = IRk \{0}/G2 . Furthermore, Cauchy and Green k kernels to IR \{0}/Vp , (1 ≤ p ≤ k) were constructed by means of adequate representatives of automorphic forms for the transversion group Vp . The transversion group is conjugated to the translation group Tp . While the Cauchy kernel associ(p) ated to the translation group, given by
0 , has no accumulation of singularities k within IR but at ∞, the Cauchy kernel function to the transversion group has a finite accumulation point of singularities, namely the origin. A combination of representation formulas for the Cauchy kernel derived for Tp , G1 and G2 lead then to further Cauchy kernel functions that are defined on manifolds that are constructed by factoring out a proper domain by a group that arises by forming the semi-direct product of Tp , G1 and G2 , respectively.

3.5. Some Perspectives

149

These kernel functions give then an immediate access to derive Plemelj projection formulas and explicit formulas for the Kerzman–Stein kernels providing important tools to study Hardy spaces on these kinds of manifolds. As mentioned in [93], also the argument principle which we described explicitly for the Euclidean space in Chapter 1.5 and for conformally flat cylinders and tori in Chapter 2.11 extends directly to the context of these manifolds when the kernel function is known. In this case one simply replaces q0 by the proper kernel functions. These results indicate a hope that it might be possible to carry out the techniques developed for these particular examples to the framework of more general discrete subgroups of the groups considered in Chapter 3.1 – Chapter 3.3. The associated s-monogenic automorphic forms would perhaps provide the building blocks for obtaining the appropriate kernel functions. Here, we see a great potential of applications of the new function classes that we constructed in this chapter. In conclusion, on the one hand, the theory of Clifford-analytic automorphic forms provides a certain potential to contribute to analytic number theory in the framework of arithmetic subgroups of the orthogonal group as a counterpart to holomorphic modular forms in several complex variables and to the non-analytic Maaß wave forms. On the other hand this theory opens the door to treat a number of problems of current interest from functional analysis, index theory, boundary value problems and spin geometry that arise in a natural way from harmonic analysis.

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List of Symbols Ak+1 Ak+1 Ak+1 (IKp,q ) B(z ∗ , R) B(z ∗ , R) B 2 (G, C), B 2 (G, Cl0k (IR)) C (C, IRk ), (C, Ak+1 ) Clp,q (IK) Cl+ p,q (IK) Cp D, DIKp,q D, DAk+1 (IKp,q ) Dhyp ∆ dσ (s) E(l) (Ωp ; Ak+1 (IKr,q )), E(l) (Ωp ) Γp (IRk ), Γp (Ak+1 ), Γp Γp [n] GV (IRk ), GV (Ak+1 ) IH H2 (G, C), H2 (G, Cl0k (IR)) H + (IRk ), H + (Ak+1 ) H r (Ak+1 ) I Im J IK L2 (·, ·) Lp M SV (Ak+1 ) IN IN0 O O Op Ωp ord(f ; c) Q

surface of (k + 1)-dim. unit ball, 18 set of paravectors in Cl0,k (IR), 3 set of paravectors in Clp,q (IK), 3 open ball with center z ∗ and radius R, 19 closed ball with center z ∗ and radius R, 19 Bergman spaces, 98 complex numbers, 2 Clifford group, 6 Clifford algebra over IKp,q , 3 even Clifford algebra, 3 cylinders, 111 Dirac operators, 16 Cauchy–Riemann operators, 16 hyperbolic Cauchy–Riemann operator, 24 Laplace operator in real Euclidean space, 23 oriented surface differential, 18 set of elliptic functions, 60 special hypercomplex modular groups, 9 principal congruence subgroups of Γp , 9 general Vahlen group 6 quaternions, 1 Hardy spaces, 98 upper half-spaces, 7 right half-space, 7 identity matrix, 9 imaginary part, 3 inversion matrix, 7 the real or complex number field, 2 L2 -spaces, 99 p-dimensional orthonormal lattice, 70 modified Vahlen group, 7 positive integers, 5 non-negative integers 5 octonions, Cayley numbers, 1 Landau symbol, 79,102 standard Z-order, 8 p-dimensional lattice, 9 order of f at c, 28 rational numbers, 8

164

List of Symbols IR Re res Res S S0 Sz ∗ Sc SV (IRk ), SV (Ak+1 ) Ta Tp Tk+1 T (Ωp ) τ (i) V Vp V ec wΓ Z

real numbers, 2 real part, 3 ordinary residue, 21 Leray–Norguet residue, 22 trace of a paravector, 4 null cone, 4 singularity cone with center z ∗ , 4 scalar part, 3 special Vahlen group 7 translation matrix, 7 translation group for Lp , 9 (k + 1)-torus, 111 translation group for Ωp , 9 a particular multi-index, 5 IRk or Ak+1 , 8 transversion group, 9 vector part, 3 winding number, 18 integers, 5

Index algebraic number fields, 84, 93 argument principle in Euclidean spaces, 33 on cylinders, 114 on tori, 114 automorphic forms, ix automorphic functions, ix Bergman kernel, 98 rectangular domains, 101, 106 wedge shaped domains, 122 Bergman projector, operator, 99 Bergman space complex, 98 hypercomplex, 100 biregular functions, 131 Brandt algebras, 84 maximal integral domains, 88 Cauchy integral formula in Euclidean spaces, 18 on cylinders, 111 on tori, 112 Cauchy integral theorem, 18 Cauchy kernel, 19 Cauchy–Riemann operator, 16 Cayley transformations, 141 class fields, x, 84 Clifford algebras, 2 Clifford conjugation, 3 Clifford main involution, 4 Clifford norm, 4 Clifford numbers, 3 Clifford reversion, 4 Clifford analysis, 16

Clifford group, 6 Clifford-analytic functions, 23 Clifford-analytic modular forms in one vector variable, 124 in two vector variables, 131 several vector variables, 142 complex conjugation, 4 complex multiplication, 83 conformal invariance formulae, 26 conformal manifolds, 111, 148 boundary value problems, 111 conformally flat cylinders, 111 conformally flat tori, 111 cusp forms one vector variable, 129 two vector variables, 140 Dirac operator, 16 Dirichlet series, 67 divisor sums, x, 72 domain manifolds, 28 Eisenstein series classical Eisenstein series, ix (p) Eisenstein series Gm,s Fourier expansion, 70 (p) Eisenstein series Gm,s , 70, 128, 134 for hypercomplex modular groups biregular Eisenstein series, 137 one vector variable, 127 two vector variables, 133 for translation groups, 49 translation + rotation invariant, 122

166

Index

elliptic functions, 57 hypercomplex division values, 93 order relations, 62 value distribution, 60 Epstein zeta function, 66, 67

meromorphic functions, 20 monogenic functions, 16 cylindrical monogenic, 111 toroidal monogenic, 111 multi-index notation, 5

Fueter polynomials, 19 Fueter–Sce solutions, 24

non-analytic automorphic forms, xi null cone, 4, 145

generalized negative powers, 19, 35 closed formulas, 47 recurrence formulas, 40, 42 generalized Riemann zeta functions, 67 Green formulas, 25 on cylinders, 112, 113

octonions, 1, 89, 90 octonionic lattices, 89 order of isolated a-points, 30, 114 orthogonal groups, xi

half-cones, 145 half-spaces, 7, 145 Hardy spaces complex, 98 hypercomplex, 100, 113 Herglotz lemma, 81 Hilbert modular forms, 131 several vector variables, 142 two vector variables, 132, 134 Hilbert’s twelfth problem, 84 holomorphic Cliffordian functions, 24 hypercomplex differentiability, 17 hypercomplex division values, 93 hypercomplex modular groups, 9, 124 congruence subgroups, 9, 124 hypercomplex multiplication, 83 hypermonogenic functions, 24 integer multiplication, 73 integer division values, 79 Laurent expansion translative Eisenstein series, 65 Laurent expansion theorem, 21 Liouville theorems, 61, 62 M¨obius transformations, 6 Maaß wave forms, xi

paravector, 3 norm, 4 trace, 4 paravector formalism, 16 paravector multiplication, 85 permutational product, 5 Poincar´e series Clifford-analytic, 130 complex-analytic, x polybiregular functions, 132 polymonogenic functions, 23 quantum gravity, x quaternionic differentiability, 11 quaternionic multiplication, 84 quaternionic division values, 85 quaternionic symplectic groups, xi quaternions, 1 rectangular domains, 101 regular points, 20 isolated a-points, 30 residue, 21 Leray–Norguet residues, 22 Riemann mapping function, 98 Riemann surfaces, x, 111 Rouch´e’s theorem in Euclidean space, 34 on cylinders, 115 s-monogenic functions, 23

Index Siegel half-space, 145 Siegel modular forms, xi Siegel type domains, 145 singular points, 20 essential singularities, 20 removable singularities, 20 unessential singularities, 20 singularity cones, 4, 50, 147 strip domains, 101 structural sets, 11 Szeg¨o kernel, 98 strip domains, 101, 108 Szeg¨o projector, 99 Taylor series expansion, 19 Theodorescu transform (periodic), 102 translation groups, 9 transversion group, 9, 148 trigonometric functions, 55 isolated zeroes, 59 Vahlen group, 6 arithmetic subgroups, 8, 124 vector formalism, 16 Weierstraß ℘-function, 57 Weierstraß ζ-function, 57 winding number, 18 Yang–Mills theory, x Z-order, 8, 85

167