Hecke’s L-functions: Spring, 1964 [1st ed. 2019] 978-981-13-9494-2, 978-981-13-9495-9

Table of contents :
Front Matter ....Pages i-xi
Algebraic Number Fields (Kenkichi Iwasawa)....Pages 1-7
Idèles (Kenkichi Iwasawa)....Pages 9-29
L-functions (Kenkichi Iwasawa)....Pages 31-66
Some Applications (Kenkichi Iwasawa)....Pages 67-92
Back Matter ....Pages 93-93

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SPRINGER BRIEFS IN MATHEMATICS

Kenkichi Iwasawa

Hecke’s L-functions Spring, 1964

SpringerBriefs in Mathematics Series Editors Nicola Bellomo, Torino, Italy Michele Benzi, Pisa, Italy Palle Jorgensen, Iowa City, USA Tatsien Li, Shanghai, China Roderick Melnik, Waterloo, Canada Otmar Scherzer, Linz, Austria Benjamin Steinberg, New York City, USA Lothar Reichel, Kent, USA Yuri Tschinkel, New York City, USA George Yin, Detroit, USA Ping Zhang, Kalamazoo, USA

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Kenkichi Iwasawa

Hecke’s L-functions Spring, 1964

123

Kenkichi Iwasawa Princeton University Princeton, NJ, USA Foreword by John Coates Emmanuel College Cambridge, UK

Masato Kurihara Department of Mathematics Keio University Yokohama, Japan

ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-981-13-9494-2 ISBN 978-981-13-9495-9 (eBook) https://doi.org/10.1007/978-981-13-9495-9 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

K. Iwasawa Princeton, 1986

Foreword

In 1964, Kenkichi Iwasawa gave a course of lectures at Princeton University on the adelic approach to Hecke’s L-functions. The present book carefully reproduces Iwasawa’s own beautifully handwritten notes used for the course, and follows faithfully his terminology. Hecke’s proof, for any number field, of the analytic continuation and functional equation of the abelian L-series, and more generally of his L-functions with Hecke characters, is of fundamental importance in algebraic number theory. Moreover, thanks to the theory of complex multiplication, it also establishes the analytic continuation and functional equation of the complex L-series of abelian varieties with complex multiplication. The modern adelic approach to Hecke’s complicated classical theory was discovered independently by Iwasawa and Tate around 1950, and marked the beginning of the whole modern adelic approach to automorphic forms and L-series. While Tate’s thesis at Princeton University in 1950 was finally published in 1967 in the volume Algebraic Number Theory edited by Cassels and Fröhlich, no detailed account of Iwasawa’s work has previously appeared, beyond a very brief note in the Proceedings of the International Congress of Mathematicians in 1950, and a short letter to Dieudonné (in Adv. Studies Pure Math. 21, 1992). The lectures presented in this volume at last provide a detailed account of Iwasawa’s work. After two preliminary chapters on the basic local and global theory of number fields, and the theory of Haar measure on the group of idèles, Chap. 3 of the book establishes the basic expression, due to Iwasawa and Tate, for the complex L-series of the Hecke L-series Lðs; vÞ attached to an arbitrary Hecke character v of a number field F as an integral over the idèle group J of F of an idelic theta function (see Sect. 3.4). Iwasawa then goes on to prove the analytic continuation and functional equation from this expression. Not only are his proofs both beautiful and fully detailed, but he also carefully explains the method in the simplest case of the Riemann zeta function. He then goes on to establish Dirichlet’s formula for the residue at s ¼ 1 of the complex zeta function of F, pointing out that an elegant

ix

x

Foreword

argument involving the compactness of the idèle class group of F also gives a non-classical proof of the finiteness of the class number and the unit theorem. In the final chapter, Iwasawa succinctly explains the link between the adelic approach and the classical theory. He then goes on to give detailed proofs of key classical results on the distribution of prime ideals, and on the class number formulae for cyclotomic fields. This volume provides an elegant and detailed account of questions which are of seminal importance for modern number theory, and it covers material which is not treated as fully or as elegantly in other basic texts on algebraic number theory. We believe that it will provide an ideal text for future courses on this central part of number theory. Finally, we warmly thank Takahiro Kitajima for his accurate conversion of Iwasawa’s handwritten notes into LATEX, and Rei Otsuki for his help with proofreading. Cambridge, UK Yokohama, Japan February 2019

John Coates Masato Kurihara

Contents

1 Algebraic Number Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Valuations (Absolute Values) and Prime Spots . . . . . . . . . . . . . . .

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3 L-functions . . . . . . . . . . . . . . . . . . . . . 3.1 Definition . . . . . . . . . . . . . . . . . . . 3.2 Theta-Formulae (Analytic Form) . . 3.3 Theta-Formulae (Arithmetic Form) 3.4 The Function f ða; s; xÞ . . . . . . . . . 3.5 Fundamental Theorems . . . . . . . . . 3.6 The Residue of fF ðsÞ at s ¼ 1 . . . .

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31 31 37 41 47 54 60

4 Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Hecke Characters and Ideal Characters . . . . . . . . 4.2 The Existence of Prime Ideals . . . . . . . . . . . . . . . 4.3 Dirichlet’s L-functions . . . . . . . . . . . . . . . . . . . . 4.4 The Class Number Formula for Cyclotomic Fields

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67 67 71 77 82

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

2 Idèles . . . . . . . . . . . . . . . 2.1 Adèles and Idèles . . . 2.2 Characters of R and J 2.3 Gaussian Sums . . . . . 2.4 Haar Measures . . . . .

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xi

Chapter 1

Algebraic Number Fields

Notation: Z = the ring of integers, 0, ±1, ±2, . . ., Q = the rational field, R = the real field, C = the complex field.

1.1 Ideals A complex number ξ is called an algebraic integer if Z[ξ ] is a finitely generated Z-module; this condition is equivalent to the fact that f (ξ ) = 0 for some polynomial f (X ) = X m + a1 X m−1 + · · · + am , ai ∈ Z. Let A be the set of all algebraic integers. Then A is a subring of C, and A ∩ Q = Z. Let F be a finite algebraic number field, i.e. Q ⊂ F ⊂ C, [F : Q] < ∞. In the following, such a field will be simply called a number field. Let o = A ∩ F. Then o(= 0) is a subring of F, and F is its quotient field. A finitely generated o-submodule of F is called an ideal (a fractional ideal) of F. Ideals of the ring o (in the usual sense) are ideals of F; they are called integral ideals of F. Let I denote the set of all nonzero ideals of F. For a, b ∈ I, a + b(= (a, b)) and ab are defined as usual. They are again ideals in I. A fundamental theorem states that I is a free abelian group with respect to the above multiplication, and that the set of all non-zero prime ideals of o forms a system of free generators of I. Namely, each ideal a in I can be uniquely written in the form a=



pm p , m p ∈ Z

(1.1)

p

where m p = 0 for almost all p. Furthermore, a is an integral ideal if and only if m p ≥ 0 for every p. The exponent m p in (1.1), which is uniquely determined by a, will be denoted by νp (a). © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2019 K. Iwasawa, Hecke’s L-functions, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-13-9495-9_1

1

2

1 Algebraic Number Fields

For any a = 0 in F, let (a) = ao. Then (a) is an ideal in I; it is called the principal ideal generated by a. (1) = o is of course the unit of I, and the set of all principal ideals forms a subgroup H of I. I/H is called the ideal class group of F. It is a finite group, and its order, [I : H], is called the class number of F. Let p be a prime ideal of o. Then o/p is a finite field. The number of elements in o/p, [o : p], is called the norm of p, and is denoted by N (p). Let a be any ideal in I, and let a be written in the form (1.1). We define the norm of a by N (a) =



N (p)m p .

p

Then N (ab) = N (a)N (b) for any a, b in I, and if a is integral, then N (a) = [o : a]. If a = ao, then N (a) = |N (a)| with N (a) the norm of a. An element a = 0 in F is called a unit of F if (a) = o; a = 0 is a unit if and only if both a and a −1 are in o. The set of all units of F forms a subgroup E of the multiplicative group F ∗ of F. Let W be the set of all roots of unity contained in F. Then W is a finite subgroup of E, and E/W is a free abelian group of rank r1 + r2 − 1 (see below). Let n = [F : Q]. Then there exist exactly n isomorphisms σ1 , . . . , σn from F into C. Let σ : F → C be one of them. Then σ : F → C, defined by σ (a) = σ (a) = the complex conjugate of σ (a), is also an isomorphism. We call σ real or complex according as σ = σ or σ = σ . We shall always fix the indices so that σ1 , . . . , σr1 are real, σr1 +1 , . . . , σn are complex, and σr1 +i = σr1 +r2 +i , 1 ≤ i ≤ r2 , n = r1 + 2r2 . For each a in F, let a (i) = σi (a) for i = 1, 2, . . . , n. Let α = (a1 , . . . , an ) be ( j) any ordered basis of F over Q. Define an n × n matrix Mα by Mα = (ai ). t Then Mα Mα = (T (ai a j )), where T denotes the trace map from F to Q. Put D(α) = det Mα tMα = (det Mα )2 = det(T (ai a j )). Then D(α) is a rational number, and D(α) = 0. It follows that there exists a unique basis β = (b1 , . . . , bn ) such that T (ai b j ) = δi j , i, j = 1, . . . , n, where δi j is Kronecker’s delta. The basis β is called the complementary basis to α. (α is then complementary to β.) The above equality is equivalent to Mα tMβ = I , namely, Mα = tMβ−1 , Mβ = tMα−1 . Let a be an ideal in I. Then a has a basis α = (a1 , . . . , an ) over Z, which is also a basis of F over Q. For any such basis α of a, D(α) is always the same. Hence we denote it by D(a). In particular, we put  = D(o), and call  the discriminant of F. Then D(a) = N (a)2 for any ideal a.  is a rational integer, and d = || > 1 for F = Q. Let  a denote the set of all b in F such that T (ab) is in Z for any a in an ideal a in I. Then  a is also an ideal in I. Let β be the complementary basis to a basis α of a over Z. Then β is a basis of  a over Z. Let d =  o−1 . Then d is an integral ideal,

1.1 Ideals

3

and N (d) = || = d. By the definition, d−1 =  o is the largest ideal of F such that T ( o) ⊂ Z. d is called the different of F. We have  a = a−1 d−1 for any ideal a.

1.2 Valuations (Absolute Values) and Prime Spots Let F be any field. A map v : F → R is called a valuation of F if (i) v(a) ≥ 0, a ∈ F; v(a) = 0 ⇔ a = 0, (ii) v(ab) = v(a)v(b), a, b ∈ F, (iii) v(a + b) ≤ C max(v(a), v(b)), a, b ∈ F, with some constant C > 0. Let a = 0, b = 0 in (iii). Then C ≥ 1. Let ||v|| = inf. of all C satisfying (iii). Then ||v|| ≥ 1, and iii) is still satisfied with C = ||v||. The valuation v is called archimedean or non-archimedean according as ||v|| > 1 or ||v|| = 1. Let v be a valuation of F. Then, for any real α > 0, v = vα is also a valuation of F. Two valuations of F are called equivalent if one is a power of the other: v = vα , α > 0. Each class of equivalent valuations of F is called a prime spot1 (or spot, or prime divisor) of F. Two equivalent valuations of F are either both archimedean or both non-archimedean. Hence we call a prime spot of F archimedean or nonarchimedean according as it consists of archimedean or non-archimedean valuations. Let v(0) = 0, v(a) = 1 for a = 0 in F. Then v is a valuation of F, and it alone constitutes a non-archimedean prime spot of F, called the trivial prime spot of F. Let v be a valuation of F. For each α > 0, let Uα denote the subset of all (a, b) in F × F such that v(a − b) < α. Then {Uα } defines a uniform structure on F, and F is a topological field relative to the Hausdorff topology defined by that uniform structure. Two valuations of F are equivalent if and only if they define the same uniform structure on F. Hence each prime spot P of F determines a uniform structure on F. Let FP denote the completion of F relative to that uniform structure. Then we can make FP into a topological field in a natural way so that F is a subfield of FP everywhere dense in FP . Each valuation v in P can be uniquely extended by continuity to a valuation v on FP , and these valuations v form a prime spot on FP . FP is called the P-completion of F. Let K be a subfield of F. The restriction of a valuation v : F → R on the subfield K defines a valuation w : K → R. Let P be a prime spot of F. Then the restrictions of valuations in P on the subfield K fulfill a prime spot Q of K . We call Q the restriction of P on K , and P an extension of Q on F. Q is archimedean or nonarchimedean according as P is archimedean or non-archimedean. The Q-completion K Q of K may be identified with the closure of K in the P-completion FP of F so that FP is an extension field of K Q . Let P1 , . . . , Pm be a finite set of prime spots of F. Let vi be a valuation in Pi , and let ai be an arbitrary element of FPi , i = 1, . . . , m. Then for any small ε > 0, there exists an element a in F such that 1 The

modern terminology of a “prime spot” is a “place”.

4

1 Algebraic Number Fields

vi (a − ai ) < ε, i = 1, . . . , m. This is called the approximation theorem. Now let F be a number field. We denote by S (resp. S0 , S∞ ) the set of all nontrivial spots (resp. non-archimedean spots, archimedean spots) of F. Spots in S0 are also called finite spots, and spots in S∞ infinite spots. Let σ1 , . . . , σn be as in Sect. 1.1, n = [F : Q]. Let ⎧ ⎨|σi (a)| (1 ≤ i ≤ r1 ), v∞,i (a) = ⎩|σ (a)|2 = |σ (a)σ (a)| (r + 1 ≤ i ≤ r + r ) i i i+r2 1 1 2 for any a in F. Then v∞,i are archimedean valuations of F, inequivalent to each other. Let P∞,i denote the prime spot of v∞,i . Then S∞ = {P∞,1 , . . . , P∞,r1 +r2 }. P∞,i is called real or complex according as σi is real or complex. σi can be extended to an isomorphism of FP∞,i onto R or C according as σi is real or complex. The valuation v∞,i will be also denoted by v P∞,i . Let p be a prime ideal of o. Define vp : F → R by vp (a) = N (p)−νp (a) , a ∈ F, where νp (a) = +∞ for a = 0, and νp (a) = νp ((a)) for a = 0. Then vp is a nontrivial non-archimedean valuation of F, and we denote the prime spot of vp by Pp . For p = p , we have Pp = Pp , and when p ranges over all prime ideals of o, S0 = {Pp }. Thus there is a one-one correspondence P ↔ p between the set S0 and the set of all prime ideals of o. The prime ideal corresponding to a prime spot P will be denoted by p P . We also write v P , ν P , and Fp for vp , νp and FP respectively if P ↔ p. Now, v P is defined for each P in S. For any a = 0 in F, v P (a) = 1 for almost all P in S, and  v P (a) = 1. P∈S

This is called the product formula. Let P1 , . . . , Pm be any finite subset of S0 . Suppose that for each i with 1 ≤ i ≤ m, an element ai in FPi and an integer m i ≥ 0 are given. Then there exists an element a in F such that

1.2 Valuations (Absolute Values) and Prime Spots

5

ν Pi (a − ai ) ≥ m i ,

i = 1, . . . , m,

ν P (a) ≥ 0,

P ∈ S0 , P = P1 , . . . , Pm .

This is called the strong approximation theorem. The restriction of a non-trivial prime spot of F on the subfield Q is a non-trivial prime spot of Q. Hence prime spots in S are obtained by extending non-trivial prime spots of Q on F. The rational field Q has a unique infinite prime spot P∞ (r1 = 1, r2 = 0) and the P∞ -completion of Q is R. The extensions of P∞ on F give us all infinite spots of F, P∞,i , 1 ≤ i ≤ r1 + r2 . A prime number p defines a prime ideal pZ of Z, and hence a finite prime spot Pp of Q. The Pp , p = 2, 3, 5, 7, 11, . . ., gives us all non-trivial finite spots g of Q. The Pp -completion of Q is the p-adic number field Q p . Let ( p) = po = i=1 piei be the prime ideal decomposition of ( p), pi = p j (i = j), ei ≥ 1. Then Ppi , i = 1, . . . , g, give us all extensions of Pp on F. Let Q be a non-trivial prime spot of Q, finite or infinite, and let P1 , . . . , Pg be the extensions of Q on F. Then there exists an isomorphism of Q Q -algebras F ⊗Q Q Q ∼ = FP1 ⊕ · · · ⊕ FPg . Let P be any one of P1 , . . . , Pg . Let n P = [FP : Q Q ], and let TP and N P denote the trace and the norm map from FP to Q Q . Then it follows from the above that n=

g  i=1

n Pi , T (a) =

g 

TPi (a),

N (a) =

i=1

g 

N Pi (a)

i=1

for any a in F. Let p be a prime ideal of o. The extensions of the functions vp and νp or Fp (= FP , P = Pp ) will be denoted again by the same letters. Let op = {a | a ∈ Fp , νp (a) ≥ 0} mp = {a | a ∈ Fp , νp (a) > 0}. Then op is a subring of Fp , mp is the unique maximal ideal of op , op = o + mp , p = o ∩ mp so that op /mp ∼ = o/p. Elements of op are called p-adic integers. Fp is a locally compact topological field, and both op and mp are open and compact in Fp . A finitely generated op -module in Fp is called an ideal of Fp . Every non-zero ideal a of Fp is principal and is a power of mp ; a = aop = msp , s = νp (a). For such a, let  be the set of all b in Fp such that Tp (ab) is in Z p for any a in a; here p is the prime number such that Q = Pp is the restriction of P on Q, Tp = TP is the trace from FP to Q Q = Q p , and Z p is the ring of p-adic integers in Q p . Then  a is also a non-zero o−1 a = a−1 d−1 ideal of Fp . Let dp =  p . Then dp ⊂ op . We call dp the different of Fp . p , and dp = op if and only if ep (= e P ) = 1.

6

1 Algebraic Number Fields

 In general, let a be an ideal of F, a = {0}, and let a = psp , sp = νp (a). Let s ap be the closure of a in Fp (a ⊂ F ⊂ Fp ). Then ap = mpp = {a | a ∈ Fp , νp (a) ≥ sp = νp (a)}, and (F ∩ ap ). a= p

a system of integers such that sp = 0 for almost all p. Let Conversely, let {sp } be

s ap = mpp , and let a = p (F ∩ ap ) = {a | a ∈ F, νp (a) ≥ sp for every p}. Then a is an ideal of F, a = {0} and  psp . a= p

Let d be the different of F. Then the closure  of d in Fp is nothing but the different dp of Fp defined in the above, and if d = ptp , tp = νp (d), then dp = mtp , and d=

(F ∩ dp ). p

Since tp = νp (d) = 0 for almost all p, we have dp = op for almost all p namely, ep (= e P ) = 1 for almost all p (almost all P in S). The multiplicative group Fp∗ of the locally compact field Fp is a locally compact abelian group in the natural topology. The groups 1 + msp , s ≥ 1, are open, compact subgroups of Fp∗ , and form a base of neighborhoods of 1 in Fp∗ . Let Up (= U P ) = {a | a ∈ Fp , νp (a) = 0} = {a | a ∈ Fp , vp (a) = 1}. Then Up is also an open, compact subgroup of Fp∗ . Up contains 1 + mp , and  Up : 1 + mp = N (p) − 1. Up is the kernel of νp : Fp∗ → Z so that Fp∗ /Up ∼ = Z. Let P = P∞,i , 1 ≤ i ≤ r1 + r2 . Then we have the topological isomorphism σi : FP → R or C. Hence FP is again a locally compact field, and FP∗ , the multiplicative group of FP , is a locally compact abelian group. Let U P = {a | a ∈ FP , v P (a) = 1}. Then

{±1}, UP ∼ = C1 = {z | z ∈ C, |z| = 1} Hence U P is always a compact subgroup of FP∗ .

if σi is real, if σi is complex.

1.2 Valuations (Absolute Values) and Prime Spots

7

Remark 1.1 Let μ be a Haar measure of the additive group of FP . Then for any measurable subset A of FP , we have μ(a A) = v P (a)μ(A), a ∈ FP . This gives us a uniform, invariant definition of v P .

Chapter 2

Idèles

2.1 Adèles and Idèles Let F be a number field, and let S, S0 , S∞ , FP etc. be as before. Put  FP . R∗ = P∈S

R ∗ is the set of all vectors (. . . , a P , . . .) where the P-components a P are taken arbitrarily from FP . R ∗ is a ring in the obvious manner; it is commutative and it has the identity 1 = (. . . , 1, 1, 1, . . .). Let a be any element of F. Then R ∗ contains the vector (. . . , a, a, a, . . .). The map a → (. . . , a, a, a, . . .) is obviously an injection of F into R ∗ . Hence we may identify a with (. . . , a, a, a, . . .), and consider F as a subfield of R ∗ . A vector α = (a P ) in R ∗ is called an adèle of F if v P (a P ) ≤ 1 for almost all P in S, namely, if a P is in o P for almost all P in S0 . The set of all adèles of F forms a subring R of R ∗ , called the adèle ring of F. If a is in F, a = 0, then v P (a) = 1 for almost all P in S. Hence a is in R. F is therefore a subfield of R. Let   oP × FP . R = P∈S0

P∈S∞

Then R is a subring of R. R is a topological ring relative to the topology which is the product of the natural topologies on o P , P ∈ S0 , and on FP , P ∈ S∞ . Since o P ’s are compact and FP ’s are finite in number, R is locally compact. Let {O} be the family of all neighborhoods of 0 in R . We define a topology on R by taking {α + O} as a base of neighborhoods of α in R. Then R becomes a (Hausdorff) topological ring, and R an open subring of R. It follows in particular that R is also locally compact. Problem 2.1 Is it possible to make R ∗ a topological ring in a similar way? Proposition 2.1 F is discrete in R. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2019 K. Iwasawa, Hecke’s L-functions, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-13-9495-9_2

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2 Idèles

 Proof Let O = o P × O1 × · · · × Or , r = r1 + r2 , where Oi = {x | x ∈ FP∞,i , v P∞,i (x) < 1}. Then O is a neighborhood of 0 in R , and hence in R. Let a be n   in F ∩ O. Then a ∈ (F ∩ o P ) = o (cf Sect. 1.2). Hence N (a) = σi (a) is an P

i=1

algebraic integer in Q, namely, a rational integer. On the other hand, |N (a)| = n r   |σi (a)| = v P∞,i (a) < 1. Hence a = 0. It follows that F ∩ O = {0} and that i=1

i=1

F is discrete in R. Remark 2.1 R/F is compact. An invertible element of the ring R is called an idèle of F. The set of all idèles of F forms a multiplicative group J , called the idèle group of F. It is easy to see that α = (a P ) is an idèle of F if and only if a P = 0 for every P in S and v P (a P ) = 1 for almost all P in S. Clearly the multiplicative group F ∗ of the field F is a subgroup of J . Let   UP × FP∗ . J = P∈S0

P∈S∞

J is a subgroup of J . Since U P is compact and FP∗ is locally compact, J is a locally compact group as product of U P , P ∈ S0 , and FP∗ , P ∈ S∞ . We then make J into a locally compact group so that J is an open subgroup of J (see the above argument for R and R). For any subset A of J , the topology on A induced by that of R will be called the adèle topology of A, and the topology on A induced by that of J will be called the idèle topology of A. It then follows from the definition that the adèle topology and the idèle topology of J are the same. Let O be a neighborhood of 1 in J in the adèle topology. Then O = J ∩ O for some neighborhood O of 1 in R. Let O = J ∩ O. Then O is a neighborhood of 1 in J in the adèle topology, and hence in the idèle topology. Therefore O is also a neighborhood of 1 in J in the idèle topology of J . As O ⊂ O , we see that the idèle topology of J is at least as strong as the adèle topology of J . Proposition 2.2 F ∗ is discrete in J (in the idèle topology of J ). Proof F is discrete in R by Proposition 2.1. Hence F ∗ = F ∩ J is discrete in J in the adèle topology of J . Hence F ∗ is also discrete in J in the idèle topology of J because the latter is at least as strong as the former. Problem 2.2 (i) For each neighborhood O of 1 in R, let O denote the set of all α in J such that both α and α −1 are in O. Prove that {O } is a base of neighborhoods of 1 in J in the idèle topology of J . (ii) Prove that the two topologies of J are different. (iii) Let P be fixed. Prove that the map R → FP (resp. J → FP∗ ) defined by α = (. . . , a P , . . .) → a P is continuous (resp. in the idèle topology of J ).

2.1 Adèles and Idèles

11

Let U=



UP .

P∈S

Since U P is always compact, U is a compact subgroup of J , and hence of J . Let J0 = {α | α = (a P ) ∈ J, a P = 1 for all P ∈ S∞ }, J∞ = {α | α = (a P ) ∈ J, a P = 1 for all P ∈ S0 }. Then J0 and J∞ are closed subgroups of J such that J = J0 × J∞ . Put U0 = U ∩ J0 , U∞ = U ∩ J∞ . Then

U = U0 × U∞ , J = U0 × J∞ , J/J = J0 /U0 .

We define the volume V (α) of an idèle α = (a P ) by V (α) =



v P (a P ).

P

Since v P (a P ) = 1 for almost all P’s, V (α) is well defined. It is easy to see that V : J → R+ is a continuous homomorphism; here R+ denotes the multiplicative group of positive real numbers. Let J1 = ker(V ) = {α | α ∈ J, V (α) = 1}. Clearly J1 is a closed subgroup of J . For each x in R+ , let τx denote the vector (a P ) such that a P = 1 for every P in S0 , 1 and a P∞,i = σi−1 (x n ) for P∞,i in S∞ . Then τx is an idèle, and the set of all such τx , x ∈ R+ , forms a subgroup T of J . Since V (τx ) = x, the maps x → τx → x = V (τx ) define a topological isomorphism T ∼ = R+ . Furthermore, T is closed in J , and

12

2 Idèles

J = J1 × T. The product formula for F states that V (a) = Hence F ∗ ⊂ J1 ,



v P (a) = 1 for any a in F ∗ .

and F ∗ T = F ∗ × T is a closed subgroup of J . Let J = J/F ∗ , J1 = J1 /F ∗ , T = F ∗ T /F ∗ . Then

J = J1 × T , T ∼ =T ∼ = R+ ,

topologically. For α = (a P ) in J , let

ι (α) =



pνPP (a P ) .

P∈S0

Since ν P (a P ) = 0 for almost all P in S0 , the product is well defined, and ι (α) is an ideal of F. The map ι : J → I = the ideal group of F is a surjective homomorphism, and its kernel is J . The restriction of ι on J0 is also surjective, and its kernel is U0 = J ∩ J0 . Hence ι (and its restriction on J0 ) induces J/J = J0 /U0 ∼ = I. Let α = (a P ) be in J0 . Then V (α) =

 P∈S0

namely,

v P (a P ) =



N (p P )−ν P (a P ) = N



pνPP (a P )

−1

S0

V (α) = N (ι (α))−1 , α ∈ J0 .

A similar computation shows that

ι (a) = (a), a ∈ F ∗ . Hence

and we have

ι (F ∗ ) = H = the group of principal ideals of F J/F ∗ J ∼ = I/H = the ideal class group of F.

= N (ι (α))−1 ,

2.1 Adèles and Idèles

13

Let E denote as before the group of units in F. An element a in F ∗ is a unit if and only if (a) = o, namely, ι (a) = 1. Hence E = F ∗ ∩ J . Let W be the group of roots of unity in F. Let ζ be in W , and let ζ m = 1. Then v P (ζ )m = v P (ζ m ) = v P (1) = 1, and hence v P (ζ ) = 1 for every P. Therefore ζ is in F ∗ ∩ U . However, since F ∗ is discrete and U is compact, F ∗ ∩ U must be finite. Hence each ζ in F ∗ ∩ U has a finite order: ζ m = 1, m ≥ 1. It follows that W = F ∗ ∩ U. We have shown at the same time that W is a finite group. Remark 2.2 In general, we can define adèles for any algebraic variety V defined over a number field F. We shall next explain such adèles in the case where V is an affine algebraic group. A subgroup G of SL(m; F), m ≥ 1, is called algebraic if there exists a set of polynomials f 1 , . . . , f s in F[X 11 , . . . , X i j , . . . , X mm ] such that G consists of all m × m matrices A = (ai j ) over F, satisfying f k (· · · , ai j , . . .) = 0, for k = 1, . . . , s. It is known in the theory of algebraic groups that an affine algebraic group defined over F is always isomorphic to such a subgroup of SL(m; F) for some m ≥ 1. Let M(m; R) denote the ring of all n × n matrices over the adèle ring R of F; it is a locally compact ring in the obvious manner. A matrix A = (αi j ) in M(m; R) is called an adèle of G if f k (. . . , αi j , . . .) = 0 for k = 1, . . . , s. The set of all such  relative to the topology adèles of G forms a multiplicative locally compact group G  the adèle group of G. The group G itself is induced by that of M(m; R). We call G  then a discrete subgroup of G.  Example 2.1 1. G = SL(m; F). In this case,  G = SL(m; R). 1a  is the group of 2. Let G be the group of all matrices with a in F. Then G 01  1α ∼ all matrices with arbitrary α in R, and G = R topologically. 01  a0  is 3. Let G be the group of all matrices with a, b in F, ab = 1. Then G 0b  α 0 the group of all matrices with α, β in R, αβ = 1. We see immediately (use 0β  α 0  onto J : G ∼ Problem) that → α is a topological isomorphism of G = J. 0β

14

2 Idèles

2.2 Characters of R and J A character of a topological group G is a continuous homomorphism χ : G → C1 . We shall next consider characters of (the additive group of) R and J . The following lemmas are easy to prove. Lemma 2.1 For any x in Q p , p = prime, there exists a unique rational number r such that (i) the denominator of r is a power of p, (ii) 0 ≤ r < 1, (iii) x ≡ r mod Zp. We shall denote such an r by x p . Lemma 2.2 (i) x p = 0 (x ∈ Q p ) ⇔ x ∈ Z p , (ii) x + y p ≡ x p + y p mod Z, for any x, y ∈ Q p , (iii) Let x be a rational number. Then x p = 0 for almost all primes p, and

x p ≡ x mod Z. p

For each P in S, we shall next define a character ω P of FP , the additive group of FP . Let P be infinite. Then put ω P (a) = exp (−2πi TP (a)) , a ∈ FP , where exp(z) = e z and TP : FP → R is, as before, the trace map. ω P is obviously a character of FP . Since (cf. Sect. 1.2) T (a) =



TP (a), a ∈ F,

P∈S∞

we obtain



ω P (a) = exp(−2πi T (a)), a ∈ F.

P∈S∞

Now, let P be finite, and let p(= Pp ) be the restriction of P on Q. Denoting again by TP : FP → Q P the trace map, we put ω P (a) = exp(2πiTP (a) p ), a ∈ FP . Using Lemma 2.2, we see that ω P (a + b) = ω P (a)ω P (b), a, b ∈ FP . Let a be in  d−1 P . Then T P (a) is in Z p so that T P (a) p = 0 and ω P (a) = 1. In particular, P =o ω P (a) = 1 for any a in o P . Hence ω P is continuous. Therefore it is a character of FP . When P ranges over all extensions of p on F, we have (cf. Sect. 1.2) T (a) =

P ext. of p

TP (a), a ∈ F.

2.2 Characters of R and J

15

By Lemma 2.2, we then obtain that 

ω P (a) = exp(2πiT (a) p ), a ∈ F.

P ext. of p

For α = (a P ) in R, let ω(α) =



ω P (a P ).

P∈S

Since a P ∈ o P and ω P (a P ) = 1 for almost all P in S0 , the product is well defined, If a P ∈ o P for and ω(α) is a number in C1 . It is clear that ω(α + β) = ω(α)ω(β). every P in S0 ; and if a P is near to 0 for every P in S∞ , then ω(α) = P∈S∞ ω P (a P ) is near to 1. Hence ω is continuous, and ω is a character of R. Let a be in F. Then ⎞ ⎛     ⎝ ω(a) = ω P (a) = ω P (a)⎠ ω P (a) P∈S

=

 p

p

P ext. of p

P∈S∞

  exp 2πiT (a) p · exp (−2πi T (a)) 

 

= exp 2πi T (a) p − T (a) p

= 1, by Lemma 2.2. So, we have the formula ω(a) = 1, a ∈ F. A character ψ of R such that ψ(F) = 1 is called a differential of F, because such characters corresponds to the differentials of an algebraic function field in the analogy between number fields and function fields. The above ω is called the fundamental differential of F. Let  ω P (a P ), ω0 (α) = P∈S0

for α = (a P ) in R. Clearly ω0 is also a character of R. Since ω(a) = 1 for a in F, ω0 (a) =

 P∈S0

 ω P (a) =



−1 ω P (a)

= exp(2πi T (a)),

P∈S∞

namely, ω0 (a) = exp(2πi T (a)), a ∈ F.

16

2 Idèles

We shall next consider a character χ of J , χ : J → C1 . Let χ0 : J0 → C1 and χ∞ : J∞ → gC1 be the restrictions of χ on J0 and J∞ respectively. For each integral ideal f = i=1 piei , ei ≥ 1 (pi prime ideal of o), put U0 (f) =

g 

(1 + mePii ) ×



U P × 1 × · · · × 1.

P= Pi P∈S0

i=1

Then U0 (f) is an open, compact subgroup of J0 , and U0 (f1 + f2 ) = U0 (f1 )U0 (f2 ), for any integral ideals f1 , f2 . For each P ∈ S0 , the groups 1 + msP , s = 1, 2, . . ., form a base of neighborhoods of 1 in FP∗ . Hence the groups U0 (f) form a base of neighborhoods of 1 in J0 , when f ranges over all integral ideals of F. Let O be a neighborhood of 1 in C1 which contains no subgroup = 1. Since χ0 is continuous, there exists an f1 such that χ0 (U0 (f1 )) ⊂ O. By the choice of O, we then have χ0 (U0 (f1 )) = 1. Suppose that χ0 (U0 (f2 )) = 1 for another integral ideal f2 . Then χ0 (U0 (f1 + f2 )) = χ0 (U0 (f1 ))χ0 (U0 (f2 )) = 1. Hence there exists a unique maximal integral ideal f such that χ0 (U0 (f)) = 1. We call f the conductor of χ0 (and of χ ). Next, we consider χ∞ . A non-zero complex number x can be written in the form x=

x · |x|. |x|

This gives us the direct decompositions R∗ = {±1} × R+ , C∗ = C1 × R+ . The characters of {±1}, C1 , and R+ (∼ = R) can be easily determined. We then see that each character ψ of R∗ is uniquely expressed in the form  ψ(x) =

x |x|

m |x|is ,

with m = 0, 1 and s ∈ R, and conversely that any such ψ defines a character of R∗ . We also have a similar result for the characters of C∗ with m = 0, ±1, ±2, . . .. The isomorphisms σk : FP∞,k → R or C induce topological isomorphisms FP∗∞,k→ ∗ R or C∗ for 1 ≤ k ≤ r . Let α = (a P ) be in J∞ , and let σ (α) = (x1 , . . . , xr ), xk = σk (a P∞,k ), 1 ≤ k ≤ r. Then σ : J∞ → R∗ × · · · × R∗ × C∗ × · · · × C∗ is also a topological isomorphism. Hence it follows from the above that the character χ∞ : J∞ → C1 is of the form

2.2 Characters of R and J

17

r   xk m k χ∞ (α) = · |xk |isk , σ (α) = (x1 , . . . , xr ), |xk | k=1 where m k = 0, 1 for 1 ≤ k ≤ r1 , m k = 0, ±1, ±2, . . . for r1 < k ≤ r , and sk ∈ R for 1 ≤ k ≤ r . Let xk+r2 = xk = σk (a P∞,k ), r1 < k ≤ r, so that |xk+r2 | = |xk |,

xk+r2 = |xk+r2 |



xk |xk |

−1

, r1 < k ≤ r.

Let  

Then

m k = m k , m k ,

sk = sk ,

1 ≤ k ≤ r1 ,

sk = 21 sk , sk+r2 = 21 sk ,

r1 < k ≤ r, m k ≥ 0,

m k = 0, sk = 21 sk , m k+r2 = −m k , sk+r2 = 21 sk ,

r1 < k ≤ r, m k < 0.

mk = m k+r2 = 0,

n   xk m k |xk |isk , χ∞ (α) = |x | k k=1

where m k = 0, 1 for 1 ≤ k ≤ r1 , m k , m k+r2 = 0, 1, 2, . . ., and m k m k+r2 = 0 for r1 < k ≤ r , sk ∈ R for 1 ≤ k ≤ n, and sk = sk+r2 for r1 < k ≤ r . We call {m 1 , . . . , m n ; s1 , . . . , sn } the signature of χ∞ (and of χ ). It determines the character χ∞ , and is uniquely determined for χ∞ by the above properties. A character χ : J → C1 such that χ (F ∗ × T ) = 1 is called a “Grössen” character of Hecke. We shall simply call it a Hecke character of F. Since J = J1 × T , J/(F ∗ × T) ∼ = J1 /F ∗ = J1 , Hecke characters of F are nothing but the characters of J1 . Let f be an integral ideal of F. Let I(f) denote  the set of all ideals a in I which are g prime to f. Then I(f) is a subgroup of I. Let f = i=1 piei , ei ≥ 1 (pi prime ideals of o, distinct), and let Pi = Ppi , i = 1, . . . , g. Put J0 (f) = {α | α = (a P ) ∈ J0 , a Pi ∈ 1 + mePii , 1 ≤ i ≤ g} = {α | α = (a P ) ∈ J0 , a Pi ≡ 1 mod mePii , 1 ≤ i ≤ g}. Then J0 ( f ) is an open subgroup of J0 such that J0 (f) ∩ U = J0 (f) ∩ U0 = U0 (f).

18

2 Idèles

The map ι0 : J0 → I induces a surjective homomorphism J0 (f) → I(f) with kernel J0 (f) ∩ U0 = U0 (f). Hence we have J0 (f)/U0 (f) ∼ = I(f). Now, let χ be a character of J with conductor f. Since χ (U0 (f)) = 1, there exists a homomorphism χ : I(f) → C1 such that χ

J0 (f)

C1

ι0

χ

I(f) is commutative. We always consider the ideal groups I, I(f), etc., in discrete topology. So χ : I(f) → C1 is a character of I(f). We call it the ideal character (of I(f)) induced by the idèle character χ . If there is no risk of confusion, we shall write χ (a) for χ (a), a ∈ I(f). By the definition, χ (a) = χ (α), a ∈ I(f), for any idèle α in J0 (f) such that a = ι (α). The relation between idèle characters and ideal characters will be studied in detail later on.

2.3 Gaussian Sums Let R be a finite ring with identity 1; for simplicity, we shall assume that R is commutative although this is not essential in the following theory. Let R∗ denote the multiplicative group of all invertible elements in R. Let λ be a character of the additive group of R, and ρ a character of the multiplicative group R∗ . Put G(λ, ρ) =



λ(x)ρ(x).

x∈R∗

Such a sum, defined for each pair of characters λ and ρ, is called a Gaussian sum on R. Example 2.2 Let R = Z/ pZ = {0, 1, . . . , p − 1 mod p}, R∗ = {1, . . . , p − 1 mod p}. Let λ(x) = exp( 2πip x ), ρ(x) = ( xp ) (Legendre’s symbol). Then G(λ, ρ) =

p−1 

x x=1

p

e

2πi x p

.

2.3 Gaussian Sums

19

For any a in R, let λa (x) = λ(ax), x ∈ R. Then λa : R → C1 is a character. Let Ra∗ = {u | u ∈ R∗ , au = a}. Then Ra∗ is a subgroup of R∗ . Lemma 2.3 (1) G(λa , ρ) = ρ(a)G(λ, ρ), if a ∈ R∗ . (2) G(λa , ρ) = 0, if ρ(Ra∗ ) = 1.

  have G(λa , ρ) = x∈R∗ λa (x)ρ(x) = λ(ax)ρ(x) Proof ∈ R∗ , then we   (1) If a −1 ρ(a)−1 λ(x)ρ(x) = ρ(a)G(λ, ρ). = λ(x)ρ(a x) = h ∗ (2) Let R = i=1 Ra∗ xi , h = [R∗ : Ra∗ ] so that x = uxi , u ∈ Ra∗ , 1 ≤ i ≤ h, h   for any x in R∗ . Then G(λa , ρ) = i=1 ∗ λ(aux i )ρ(ux i ) = u∈R i λ(ax i )ρ(x i ) a   ∗ ρ(u). However, since ρ(R )  = 1, ρ(u) = 0. Hence G(λ , ρ) = 0. a a u u Lemma 2.4 Suppose that

(i) λa (R) = 1 for any a in R, a = 0, and / R∗ . (ii) ρ(Ra∗ ) = 1 for any a in R, a ∈ Then |G(λ, ρ)|2 = #(R). In particular, G(λ, ρ) = 0.    Proof |G(λ, ρ)|2 = x∈R∗ λ(x)ρ(x) y∈R∗ λ(y)ρ(y) = x,y λ(x − y)ρ( xy ). Let z = xy . Then the sum is equal to ⎛

λ((z − 1)y)ρ(z) =

y,z∈R∗

However,



=

⎜ ⎟ ⎜ ⎟. − ⎝ ⎠

z∈R∗



z∈R∗ y∈R

ρ(z)

z



⎞

y∈R

y∈R y ∈R / ∗

λz−1 (y) = ρ(1)#(R) = #(R),

y∈R

because λz−1 ≡ 1 for z = 1. On the other hand,



z∈R∗ y ∈R / ∗

=



λ(−y)

y ∈R / ∗



λ(zy)ρ(z) =

z∈R∗



λ(−y)G(λ y , ρ).

y ∈R / ∗

By the assumption (ii) and by Lemma 2.3 (1), we have G(λ y , ρ) = 0 for every y in R, y ∈ / R∗ . Hence the above sum is 0, and the lemma is proved. Example 2.3 For G(λ, ρ) = λ, ρ. Hence

 p−1

2πi x x p x=1 ( p )e

, the assumptions (i), (ii) are satisfied for

|G(λ, ρ)|2 = #(R) = p.

20

2 Idèles

Let a be an ideal of F, a = {0}, a = Ra =



ν (a)

mpp

p

×





pνp (a) . Put

FP

P∈S∞

= {α | α = (a P ), a P ∈ mνPP (a) , for all P ∈ S0 } = {α | α = (a P ), ν P (a P ) ≥ ν P (a), for all P ∈ S0 }.   For a = o, Ra = op × FP = R (in Sect. 2.1). In general, Ra is an open R submodule of R, and F ∩ Ra = a. If a ⊂ a , then Ra ⊂ Ra . Let a be integral. Then Ra ⊂ R = Ro , and Ra is an ideal of R . Let Ra = R /Ra , a ⊂ o. Proposition 2.3 (1) There is a natural isomorphism Ra ∼ = o/a, so that Ra is a finite ring with N (a) elements. (2) Let α = (a P ) be an element of R , and let α be the residue class of α in Ra = R /Ra . Then α is in R∗a if and only if ν P (a P ) = 0 for every P ∈ S0 such that ν P (a) > 0. Proof (1) a = o ∩ a = o ∩ F ∩ Ra = o ∩ Ra . Let α = (a P ) be in R . By the strong approximation theorem, there exists a in F such that a P ≡ a mod mνPP (a) for every P ∈ S0 . Let β = α − a. Then β is in Ra . Hence a = α − β is in R , and consequently in o = F ∩ R . Thus α = a + β with a ∈ o, β ∈ Ra . Therefore R = o + Ra , and together with a = o ∩ Ra , we obtain Ra = R /Ra ∼ = o/a. (2) Suppose that α ∈ R∗a . Then αβ = 1, namely, αβ ≡ 1 mod Ra for some β = (b P ) in R . It follows that ν P (a P b P − 1) ≥ ν P (a) for every P ∈ S0 . Suppose that ν P (a) > 0. Then ν P (a P b P − 1) > 0 so that ν P (a P b P ) = 0. Since ν P (a P ), ν P (b P ) ≥ 0, we obtain ν P (a P )(= ν P (b P )) = 0. Suppose conversely that ν P (a P ) = 0 for every P ∈ S0 such that ν P (a) > 0. Let β = (b P ) where b P = a −1 P if P ∈ S0 and ν P (a) > 0, and b P = 0 otherwise. Then β is in R , and ν P (a P b P − 1) ≥ ν P (a) for any P ∈ S0 . Hence αβ ≡ 1 mod Ra , i.e. αβ = 1, and thus α is in R∗a . Problem 2.3 Prove that

R∗a ∼ = U0 /U0 (a), a ⊂ o.

Let a be an ideal of F, a = {0}. An element α in J is called an idèle associated with a if ι (α0 ) = a, α∞ = τ N (a) , for α = α0 α∞ , α0 ∈ J0 , α∞ ∈ J∞ . If α = (a P ), the condition is equivalent to that

2.3 Gaussian Sums

21

ν P (a P ) = ν P (a) for P ∈ S0 , a P∞,i = σi−1 (N (a) n ) for P∞,i ∈ S∞ . 1

Since ι0 : J0 → I is surjective, such an α always exists, and if α is also associated with a, then α = αξ , ξ ∈ U0 , and conversely Problem 2.4 Prove that α is associated with a if and only if

ι (α) = a, α ∈ (J0 × T ) ∩ J1 . Now, let d be as before the different of F, and let d = N (d) = |Δ|, where Δ is the discriminant of F. Let δ be an idèle associated with d, namely, δ = (d P ), 1 ν P (d P ) = ν P (d), P ∈ S0 , d P∞,i = σi−1 (d n ), P∞,i ∈ S∞ . Let d P be the different of FP , P ∈ S0 . Then d P = mνPP (d) (cf. Sect. 1.2). Hence −1 d P = d P o P , o P = dP =

1 oP , dP

P ∈ S0 .

Let χ : J → C1 be a character of J , and let f be the conductor of χ . Let ϕ be an idèle associated with f, namely, ϕ = ( f P ), ν P ( f P ) = ν P (f), P ∈ S0 , f P∞,i = 1 σi−1 ( f n ), P∞,i ∈ S∞ where f = N (f). Let R = Rf = R /Rf ∼ = o/f. Then R is a finite ring with f elements. We shall fix χ , ϕ, and δ, and define a Gaussian sum on the ring R. I. The Additive Character λ : R → C1 Let ω0 : R → C1 be the character defined in Sect. 2.2. Define ω0 : R → C1 by ω0 (ξ ) = ω0



ξ δϕ



, ξ ∈ R .

ξ Since δ, ϕ are in J , δϕ is in R. Hence ω0 is well defined, and is obviously a character of R . Let α = (a P ) be an element of R . We shall prove that ω0 (αξ ) = 1 for every ξ = (x P ) in R ⇔ α is in Rf . Suppose that ω0 (αξ ) = 1 for any ξ = (x P ) in R . Then

ω0 (αξ )

 = ω0

αξ δϕ

=

 P∈S0

 ωP

aP x P dP f P

= 1.

Let P be fixed, P ∈ S0 , and let ξ = (x p , 0, . . . , 0) with arbitrary x P in o P . Then ξ is in R , and it follows from the above that

22

2 Idèles

 ωP

aP x P dP f P









= exp 2πi T p

aP x P dP f P

  = 1, p

namely, that T p ( da PP xf PP ) is contained in Z p ( p = restriction of P on Q). Since this 1 holds for any x P in o P , we see that d Pa Pf P ∈ o P = d P o P , namely, that a P ∈ f P o P , ν P (a P ) ≥ ν P ( f P ) = ν P (f). As this is true for arbitrary P in S0 , α = (a P ) is contained in Rf . Conversely, if α = (a P ) is in Rf , then ν P (a P ) ≥ ν P (f), P ∈ S0 , and the above computation shows that ω P ( da PP xf PP ) = 1 for any x P in o P . Hence ω0 (αξ ) = 1 for any ξ = (x P ) in R . In particular, it follows from the above that ω0 (α) = 1 if α is in Rf . Hence ω0 (α) depends only upon the residue class α of α mod Rf , and we obtain a character λ : R = R /Rf → C1 by putting λ(α) = ω0 (α), α ∈ R (i.e. α ∈ R ). The above result then shows that λα (ξ ) = λ(αξ ) = ω0 (αξ ) = 1 for any ξ in R (for any ξ in R ) ⇔ α = 0. Thus λα (R) = 1, for any α in R, α = 0. II. The Multiplicative Character ρ : R∗ → C1 For any α = (a P ) in R, let αf = (a P ) be defined by a P

 a P , if P ∈ S0 and ν P (f) > 0, = 1, otherwise.

Then α → αf is a continuous map R → R such that (αβ)f = αf βf . The restriction on J defines a continuous homomorphism J → J . Note that if f = 0, then αf = 1 for any α in R. Define χf : R → C by  χf (α) =

χ (αf ), if αf ∈ U0 (or αf ∈ U ), 0, otherwise.

For example, χf (0) = 1 or 0 according as f = o or f = o. Also, if α is in U0 , then αf ∈ U0 , αf /α ∈ U0 (f) so that χf (α) = χ (αf ) = χ (α(αf /α)) = χ (α). Let α = (a P ) be in R , and let α be the residue class of α mod Rf , α ∈ R. We shall next prove that χf (α) depends only upon α. It follows from Proposition 2.3 that

2.3 Gaussian Sums

23

χf (α) = χ (αf ) or χf (α) = 0 / R∗ . Hence it is sufficient to show that χ (αf ) = χ (βf ) according as α ∈ R∗ or α ∈ if α = β, β = (b P ) ∈ R . Let ν P (f) > 0 so that ν P (a P ) = ν P (b P ) = 0. Since α ≡ β mod Rf , ν P (a P − b P ) ≥ ν P (f). Hence ν P ( ab PP − 1) = ν P (a P − b P ) − ν P (b P ) ≥ α α ν P (f) for any P ∈ S0 with ν P (f) > 0. Therefore βff is contained in U0 (f), and χ ( βff ) = 1, namely χ (αf ) = χ (βf ). We now define ρ : R∗ → C1 by ρ(α) = χ f (α) = χf (α)−1 = χ (αf )−1 , α ∈ R∗ , where α is any element of R in the residue class α. By the above remarks, ρ is well-defined, and gives a character of R∗ . / R∗ . Let α = (a P ) be in the residue class α. Then there exists a spot Let α ∈ R, α ∈ P0 ∈ S0 such that ν P0 (a P0 ) > 0, ν P0 (f) > 0 (Proposition 2.3). Let P0 ↔ p0 , f = f/p0 . Since ν P0 (f ) = ν P0 (f) − 1 ≥ 0, f is an integral ideal of F, f ⊂ f , f = f . Hence χ (U0 (f)) = 1 by the definition of the conductor f. Let ξ = (x P ) ∈ U0 (f ), χ (ξ ) = 1. Since ξ is in U0 , χf (ξ ) = χ (ξ ) = 1. Hence ξ is in R∗ , and ρ(ξ ) = χ f (ξ ) = 1. On the other hand, ν P (a P x P − a P ) = ν P (a P ) + ν(x P − 1) ≥ ν P (a P ) + ν P (f ), P ∈ S0 . If P = P0 , then ν P (f ) = ν P (f) and ν P (a P ) ≥ 0 (α ∈ R ) so that ν P (a P x P − a P ) ≥ ν P (f). If P = P0 , then ν P (f ) = ν P (f) − 1, but ν P (a P ) ≥ 1 so that again ν P (a P x P − a P ) ≥ ν P (f). Hence αξ − α is in Rf , namely, αξ = α. Therefore ξ belongs to the subgroup R∗α of R∗ , and it follows from the above that / R∗ . ρ(R∗α ) = 1, α ∈ Thus the condition (ii) of Lemma 2.4 is satisfied for the character ρ. We now put C(χ ; δ, ϕ) = G(λ, ρ)

λ(ξ )ρ(ξ ), = ξ ∈R∗

with the characters λ : R → C1 and ρ : R∗ → C1 defined in the above. Note that if f = o, then Rf = Ro = R so that R = {0}, R∗ = {1} = {0} = R, and C(χ ; δ, ϕ) = 1. Proposition 2.4  |C(χ ; δ, ϕ)| = f , f = N (f), C(χ ; δ, ϕ) = χf (−1)C(χ ; δ, ϕ). Proof The first equality follows immediately from Lemma 2.4. As to the second,

24

2 Idèles

C(χ ; δ, ϕ) = =



λ(ξ )ρ(ξ ) =



λ(−ξ )ρ(ξ )

λ(ξ )ρ(ξ ). λ(ξ )ρ(−ξ ) = ρ(−1)

Here ρ(−1) = χf (−1), ρ(ξ ) = χ f (ξ ). Hence and the equality is proved.



λ(ξ )ρ(ξ ) = G(λ, ρ) = C(χ ; δ, ϕ),

Let a and b be ideals of F, a, b = {0}, b ⊂ a, and let ξ ∈ J . Then 1. Ra = a + Rb , b = a ∩ Rb so that a/b ∼ = Ra /Rb under x + b ↔ x + Rb , x ∈ a, 2. ξ Ra = Rι (ξ )a so that Ra ∼ = Rι (ξ )a under α ↔ ξ α, α ∈ Ra . Problem 2.5 Prove the above 1, 2. We now prove the following. Proposition 2.5 Let α ∈ J , a = ι (α), a ∈ ad−1 f−1 (b ∈ F). Then

ω0 (xa)χ f (xα) = χf (α −1 δϕa)C(χ ; δ, ϕ),

x

where x ranges over a set of representatives of a−1 modulo a−1 f. Proof When x moves as stated, then x also ranges over a set of representatives of Ra−1 mod Ra−1 f (see 1. in the above). Hence ξ = xα ranges over a set of representatives of α Ra−1 = Rι (α)a−1 = Ro = R modulo α Ra−a f = Rι (α)a−1 f = Rf . Hence the left hand side of the equality in the proposition is equal to

ξ ∈R

 ω

aξ α

χ f (ξ ) =



 ω0

ξ ∈R

α ξ δϕ

χ f (ξ ),

where α = α −1 δϕa. Let α = (a P ), α = (a P ). Then a P = a −1 P d P f P a so that ν P (a P ) = −ν P (a P ) + ν P (d P ) + ν P ( f P ) + ν P (a) = −ν P (a) + ν P (d) + ν P (f) + ν P (a) = ν P (a−1 dfa) ≥ 0 because a ∈ ad−1 f−1 , aa−1 df ⊂ o. Hence α is in R . As χ f (ξ ) = 0 / R∗ , we see that the above sum is equal to if ξ ∈

ξ ∈R∗

λ(α ξ )ρ(ξ ) =



λα (ξ )ρ(ξ )

ξ

= G(λα , ρ). Suppose that α ∈ R∗ . Then it follows from Lemma 2.3 that G(λα , ρ) = ρ(α )G(λ, ρ) = χf (α )C(χ ; δ, ϕ) = χf (α −1 δϕa)C(χ ; δ, ϕ). / R∗ . Then ρ(R∗α , ρ) = 1. Hence it follows again from Lemma Suppose that α ∈ 2.3 that G(λα , ρ) = 0. However, in this case, χf (α ) = 0. Hence the proposition is proved.

2.4 Haar Measures

25

2.4 Haar Measures We shall next state without proof some fundamental results on Haar measures of locally compact groups. For simplicity, we shall always assume that our locally compact groups are abelian and separable (i.e. have a countable basis for open sets). Problem 2.6 Prove that R and J are separable. Let G be such a locally compact, separable, abelian group. A Borel (=Baire) measure μ on G is a measure defined on the family of Borel (=Baire) subsets of G such that μ(C) < +∞ whenever C is compact. A Borel measure μ on G is called a Haar measure if μ ≡ 0 and if μ(a A) = μ(A) for any a in G and for any Borel subset A of G. The fundamental theorem states that there exists such a Haar measure on G, and that it is unique up to a constant factor. 1. Let μ be a Haar measure on G. Let f (x) ∈ L 1 (G, μ). Then f (x −1 ), f (a −1 x) ∈ L (G, μ), a ∈ G, and    −1 f (x)dμ(x) = f (x )dμ(x) = f (a −1 x)dμ(x), 1

G

namely,

G

G

dμ(x) = dμ(x −1 ) = dμ(ax), a ∈ G.

2. G is compact if and only if μ(G) < +∞ for a Haar measure μ on G. In such a case, for any given c > 0, there exists a unique Haar measure μ on G such that μ(G) = c. 3. G is discrete if and only if each point of G has a positive measure for a Haar measure of G. In such a case, there exists a unique Haar measure μ on G such that μ(x) = 1 for every x in G. We shall call such μ the point measure on G. 4. Let σ : G → G be a topological isomorphism. Let μ be a Haar measure on G. Define μ by μ (A ) = μ(σ −1 (A )) for any Borel subset A of G . Then μ is a Haar measure of G . We denote it by μσ . We also write μ ≈ μ when μ and μ are σ related as mentioned in the above. 5. Let μ1 , μ2 be Haar measures on G 1 and G 2 respectively. Then the product measure μ1 × μ2 is a Haar measure on G 1 × G 2 . Let H be a closed subgroup of G. Then both H and G/H are again locally compact, separable, abelian groups. Let μG , μ H , and μG/H be Haar measures of the respective  groups. Let f be a continuous function with compact support on G. Then g(x) = H f (xu)dμ H (u) is defined for any x in G, and g(x) depends only upon the coset x = x H . Hence we put g (x ) =



f (xu)dμ H (x), x = x H ∈ G/H. H

26

2 Idèles

g is then continuous, and has a compact support in G/H . Hence The function G/H g (x )dμG/H (x ) is defined. Now, we say that the measures μG , μ H , and μG/H are matched if 



g (x )dμG/H (x ) G/H   = f (xu)dμ H (u) dμG/H (x ),

f (x)dμG (x) = G

G/H

(2.1)

H

for any function f which is continuous and has a compact support on G. Given Haar measures on any two of the groups G, H , and G/H , there exists a unique Haar measure on the third group such that the three measures thus obtained are matched in the above sense. If μG , μ H , and μG/H are matched, we write μG = μG/H ∗ μ H , μG/H = μG /μ H . 6. Let H ⊂ G, μG = μG/H ∗ μ H . Let f be a non-negative measurable function on G. Then g (x ) = H f (xu)dμ H (u) is also non-negative and measurable on G/H , and (2.1) holds. 7. In the same situation, let f be a function in L 1 (G, μG ). Then g (x ) =

 f (xu)dμ H (x) H

is defined for almost all x in G/H ; g is a function in L 1 (G/H, μG/H ), and (2.1) again holds. Let H be a closed subgroup of G; and U a closed subgroup of H . Let μG/H and μ H/U be Haar measures of the respective groups. Let G = G/U , H = H/U . Then G/H is naturally identified with G /H so that μG/H = μG /H . As μ H/U = μ H , μG/H ∗ μ H/U = μG /H ∗ μ H is defined, and it is a Haar measure on G/U . 8. Let V ⊂ U ⊂ H ⊂ G be a sequence of closed subgroups, and let μG/H , μ H/U , μU/V be given Haar measures on the respective groups. Then Associative law: μG/H ∗ (μ H/U ∗ μU/V ) = (μG/H ∗ μ H/U ) ∗ μU/V . 9. Let U ⊂ H ⊂ G, and let μG , μ H , and μU be given. Then (μG /μU )/(μ H /μU ) = μG /μ H . This follows from 7 by putting V = 1, μG/H = μG /μ H , μ H/U = μ H /μU , μU/V = μU . 10. Let H1 ⊂ G 1 , H2 ⊂ G 2 , and let μG 1 , μG 2 , μ H1 , μ H2 be given. Then (μG 1 × μG 2 )/(μ H1 × μ H2 ) ≈ (μG 1 /μ H1 ) × (μG 2 /μ H2 ) under the natural isomorphism (G 1 × G 2 )/(H1 × H2 ) ∼ = (G 1 /H1 ) × (G 2 /H2 ).

2.4 Haar Measures

27

11. Let μG 1 and μG 2 be given. Then (μG 1 × μG 2 )/μG 1 ≈ μG 2 , (μG 1 × μG 2 )/μG 2 ≈ μG 1 , under the natural isomorphisms (G 1 × G 2 )/G 1 ∼ = G 2 , (G 1 × G 2 )/G 2 ∼ = G1. We also note that given μG 1 ×G 2 and μG 1 (μG 2 ), there exists a unique μG 2 (μG 1 ) such that μG 1 ×G 2 = μG 1 × μG 2 . Example 2.4 1. G = Rn , x = (x1 , . . . , xn ) ∈ Rn , n ≥ 1. dμ(x) = d x1 · · · d xn . 2. G = R∗ or R+ , x ∈ R∗ or R+ dμ(x) =

dx . |x|

3. G = C∗ , z ∈ C∗ , z = x + i y = r eiθ . dμ(z) =

2d xd y 2dr dθ . = 2 |z| r

The Haar measures of Rn , R∗ , R+ , and C∗ as normalized in the above, will be called the standard measures on these groups. We shall next fix Haar measures of various groups defined by J and its subgroups. Let μU0 be the Haar measure of the compact group U0 such that μU0 (U0 ) = 1, and let μ J0 /U0 be the point measure of the discrete group J0 /U0 . Let μ J0 = μ J0 /U0 ∗ μU0 . Let σ : J∞ → R∗r1 × C∗r2 , σ (α) = (x1 , . . . , xr ), r = r1 + r2 , be the topological isomorphism defined in §2.2. Let μ be the Haar measure of R∗r1 × C∗r2 which is the product of the standard measures on the factors. Let μ J∞ ≈ μ , under σ . Let μ J = μ J0 × μ J∞ . Thus the Haar measure μ J on J is fixed. Let μR+ be the standard measure on R+ , and let μT ≈ μR+

28

2 Idèles

under the isomorphism T → R+ , τx → x. Then there exists a unique Haar measure μ J1 on J1 such that μ J = μ J1 × μT . Let μ F ∗ be the point measure on the discrete group F ∗ , and put μ J = μ J /μ F ∗ , μ J1 = μ J1 /μ F ∗ . Let μT ≈ μT under T = (F ∗ × T )/F ∗ → T . Then μ J1 × μT ≈ μ J1 × μT under the isomorphism J = J1 × T → J1 × T induced by T → T . However, the above isomorphism can be written in the form J = (J1 × T )/(F ∗ × 1) → (J1 /F ∗ ) × (T /1). Let μ1 be the point measure of 1. Then μ F ∗ × μ1 = μ F ∗ , μT /μ1 = μT . By 10 in the above, we have (μ J1 × μT )/(μ F ∗ × μ1 ) ≈ (μ J1 /μ F ∗ ) × (μT /μ1 ), under J → (J1 /F ∗ ) × (T /1), namely, μ J = μ J /μ F ∗ ≈ μ J1 × μT . Therefore μ J = μ J1 × μT . Remark 2.3 Let μ be any Haar measure of the additive group R. Let ξ be an element −1 of J . Then σξ : R → R, α → ξ α, is a topological isomorphism. Let μ = μσξ so that μ (A) = μ(σξ (A)) = μ(ξ A) for any Borel set A in R. Then μ is also a Haar measure of R, and μ = cξ μ with some constant cξ > 0. It can be proved that cξ = V (ξ ), namely, that μ(ξ A) = V (ξ )μ(A), ξ ∈ J, A ⊂ R. This gives us an invariant definition of the volume V (ξ ).

2.4 Haar Measures

29

Let a = ξ be in F ∗ . Then σa (F) = F so that σa induces an isomorphism σa : R/F → R/F. Let  μ be a Haar measure of R/F. Then we see easily that  μ(σa (A )) = μ(A ) for any Borel set A in R/F, with the same ca = V (a). Let A = R/F. ca  Then σa (A ) = A . As R/F is compact, 0 <  μ(A ) < +∞. Hence it follows from the above that ca = 1, namely, that V (a) = 1, a ∈ F ∗ . Thus the product formula for F is essentially a consequence of the compactness of R/F.

Chapter 3

L-functions

3.1 Definition Let D be a domain in the complex s-plane. Let u 1 (s), u 2 (s), . . . , be a sequence of ∞holomorphic functions on D such that u m (s) = 1, s ∈ D, m ≥ 1, and that m=1 |u m (s)| converges uniformly on D to a bounded function on D. Then lim

N →∞

N 

(1 − u m (s)), s ∈ D,

m=1

converges uniformly to a holomorphic function u(s) on D such that u(s) = 0, s ∈ D. N  Furthermore, if { j1 , j2 , . . .} is any permutation of {1, 2, . . .}, then lim (1 − N →∞

m=1

u jm (s)) converges to the same u(s) in a similar way. The function u(s) is denoted by ∞ 

(1 − u m (s)), s ∈ D.

m=1

Now, let F be as before a number field, and let χ : J → C1 be a Hecke character of F; χ (F ∗ × T ) = 1. Let f be the conductor of χ , and let χ (= χ  ) : If → C1 be the character of the ideal group If induced by χ : J → C1 . For any integral ideal a which is not prime to f, put χ (a) = 0. Then χ (a) is defined for every integral ideal of F, and χ (ab) = χ (a)χ (b) for any integral ideal a and b. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2019 K. Iwasawa, Hecke’s L-functions, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-13-9495-9_3

31

32

3 L-functions

Let p be a prime ideal of o ⊂ F. We write p| p if p(= Pp ) is the restriction of Pp on Q. In such a case, N (p) = p f , f ≥ 1. Consider the function N (p)N (p)−s which is holomorphic on the entire s-plane. Since |χ (p)N (p)−s | = |χ (p)||N (p)−s | ≤ N (p)− Re (s) we have

|χ (p)N (p)−s | < 1, χ (p)N (p)−s = 1, for Re (s) > 0.

Let

1 − u p (s) = (1 − χ (p)N (p)−s )−1 ,

namely, u p (s) =

−χ (p)N (p)−s , Re (s) > 0. 1 − χ (p)N (p)−s

Then u p (s) is clearly holomorphic, and u p (s) = 1 for Re (s) > 0. Let a > 1 and let Da = {s | Re (s) > a}. For s in Da , |x(p)N (p)| ≤ N (p)− Re (s) ≤ N (p)−a = p −a f ≤ p −a ≤ 2−a ≤ 21 . Hence |u p (s)| ≤

p −a = 2 p −a , s ∈ Da , 1 − 21

For each prime number p, there exist at most n = [F : Q] prime ideals p such that p| p (cf. I. §2). Hence, when p ranges over all prime ideals of o, we obtain 

⎛ ⎛ ⎞ ⎞     ⎝ ⎝ |u p (s)| = |u p (s)|⎠ ≤ 2 p −a ⎠

p

p

≤



p| p

2np −a = 2n

p



p

p| p

p −a < +∞, s ∈ Da .

p

 This shows that p |u p (s)| converges uniformly to a bounded function on Da (and also that {u p (s)} is a countable set). Applying the result stated at the beginning to D = Da , we obtain a holomorphic function 

(1 − u p (s)) =

p

 (1 − χ (p)N (p)−s )−1 p

on Da . Since this holds for any a > 1, we see that the same product defines a function of s for any s with Re (s) > 1. So, let L(s; χ ) =

 p

(1 − χ (p)N (p)−s )−1 , for Re (s) > 1.

3.1 Definition

33

L(s; χ ) is called Hecke’s L-function for F with the character χ , as it follows from the above, L(s, χ ) is holomorphic and L(s; χ ) = 0 for Re (s) > 1, and m 

(1 − χ (pk )N (pk )−s )−1

k=1

converges uniformly to L(s; χ ) in Da = {s | Re (s) > a}, a > 1, for any ordering p1 , p2 , p3 , . . . , of the prime ideals of o. Let χ ≡ 1. Then f = o, and χ (a) = 1 for any ideal a of F. In this case, we write ζ F (s) for L(s; χ );

ζ F (s) =

 (1 − N (p−s )−1 , Re (s) > 1. p

ζ F (s) is called Dedekind’s zeta-function for the number field F. Proposition 3.1 When a ranges over all integral ideals of F, the Dirichlet series 

χ (a)N (a)−s

a

converges absolutely and uniformly on each domain Da , a > 1, and L(s; χ ) =



χ (a)N (a)−s , for Re (s) > 1.

a

Proof For any real x > 0, there exist only a finite number of prime ideals p such that N (p) ≤ x. We have 

(1 − N (p)−a )−1 =

N (p)≤x



(1 + N (p)−a + N (p)−2a + · · · )

N (p)≤x

=





N (a)−a ,

a

where the sum is taken over all integral ideals a such that the prime factors p satisfy N (p) ≤ x. Clearly every integral ideal a with N (a) ≤ x is included in the above sum. Hence     N (a)−a ≤ N (a)−a = (1 − N (p)−a )−1 a

N (a)≤x a⊂o

≤

 p

N (p)≤x

(1 − N (p)−a )−1 = ζ F (a).

34

3 L-functions

Since this holds for any x > 0, it follows that 

N (a)−a ≤ ζ F (a) < +∞.

a⊂o

|χ (a)N (a)−s | ≤ N (a)− Re (s) ≤ N (a)−a , we see from Now, let s be in Da . Since −s the above that χ (a)N (a) converges absolutely and uniformly in Da . A computation similar to the above also shows that    (1 − χ (p)N (p)−s )−1 = χ (a)N (a)−s , s ∈ Da , a

N (p≤x)

because χ (ab) = χ (a)χ (b), a, b ⊂ o. Hence   −s −1 −s (1 − χ (p)N (p) ) − χ (a)N (a) N (p)≤x N (a)≤x   = N (a)−s N (a)>x    ≤ N (a)−a ≤ N (a)−a . N (a)>x

N (a)>x

  −a −a Let x → +∞. Since converges, → 0. Hence we a N (a) N (a)>x N (a) obtain   (1 − χ (p)N (p)−s )−1 = χ (a)N (a)−s , s ∈ Da . L(s; χ ) = p

a

Since a(>1) is arbitrary, the above equality also holds for any s with Re (s) > 1. For χ ≡ 1, we have

ζ F (s) =



(1 − N (p)−s )−1 =

p



N (a)−s , Re (s) > 1.

a

Let F = Q. Then ζ (s) = ζQ (s) is the well-known zeta-function of Riemann;

ζ (s) =



(1 − p −s )−1 =

p

∞ 

m −s , Re (s) > 1.

m=1

For ζ (s), Riemann proved the following. Let s > 1. We have Γ

s  2

= 0

∞

e−x x 2 s

dx . x

3.1 Definition

35

Hence π−2 Γ s

s  2

m −s =



∞

e−πm x x 2 s

2

0

dx . x

It follows that π

− 2s

Γ

s  2

ζ (s) =

∞ 

∞

m=1 0 ∞ ∞

=



0

0

where ω(x) =

∞ 

dx x

e−πm x x 2

dx x

2

2

m=1 ∞

=

e−πm x x 2

s

ω(x)x 2

s

s

dx , x

e−πm x , x > 0. 2

m=1

The interchange of the sum and the integral is allowed because e−πm x x 2 ≥ 0. Let  2 θ (x) = e−πm x = 1 + 2ω(x), x > 0. 2

m∈Z

Then the classical theta-formula states that

 1 1 , x > 0, θ (x) = √ θ x x namely, that

 1 1 1 1 ω(x) = √ ω − + √ , x > 0. x 2 2 x x

Hence we have ∞

1 ∞ dx s dx s dx = + , ω(x)x 2 ω(x)x 2 x x x 0 0 1 

 1 1 1 1 1 1 s dx s dx x2 = − + √ , ω(x)x 2 √ ω x x 2 2 x x x 0 0

0

1

s

ω(x)x 2

 1

 s s−1 1 s 1x2 1 s−1 d x 1x 2 − x2 + x 2 = − s + s−1 , s>1 2 2 x 2 2 2 2 0

1 1 =− + s s−1 1 , = s(s − 1)

s

36

3 L-functions

so that 1

 1 1 s−1 d x + x 2 x x s(s − 1) 0 0 ∞ 1 1−s d x + , ω(x)x 2 = x s(s − 1) 1 ∞ ∞

s  1 s s dx 1−s d x π−2 Γ + + , s > 1. ζ (s) = ω(x)x 2 ω(x)x 2 2 x x s(s − 1) 1 1 s

ω(x)x 2

dx = x



1

ω

This shows in particular that

∞

s

ω(x)x 2

1

dx < +∞ x

for any s > 1. s Let a > 1, and let s be any complex number such that Re (s) < a. Then |x 2 | = s a x Re ( 2 ) ≤ x 2 for x ≥ 1. Hence ∞ ∞ s dx a dx 2 ≤ < +∞. |ω(x)x | ω(x)x 2 x x 1 1

This shows that

∞

η(s) =

s

ω(x)x 2

1

dx x

defines a holomorphic function of s for Re (s) < a. Since a (> 1) is arbitrary, η(s) actually defines a holomorphic function on the entire s-plane. Let

s  s ξ(s) = π − 2 Γ ζ (s), Re (s) > 1. 2 It follows from the above that ξ(s) = η(s) + η(1 − s) +

1 , for s > 1. s(s − 1)

(3.1)

Since both sides of the above are holomorphic function of s for Re (s) > 1, the equality holds for any s with Re (s) > 1. Moreover, the right-hand side is a meromorphic function on the entire s-plane with the only simple poles at s = 0, 1. Hence, by the above equality ξ(s) can be continuous analytically on the entire s-plane, and so is  −1  −1 s ζ (s) = π 2 Γ 2s ξ(s). Since Γ 2s has a zero at s = 0, ζ (s) is regular at s = 0. 1 1 Since Γ 2 = π 2 and since ξ(s) has a simple pole with residue 1 at s = 1, ζ (s) also has a simple pole with residue 1 at s = 1. Furthermore, it follows from (3.1) (which now holds for arbitrary s) that

3.1 Definition

37

ξ(s) = ξ(1 − s). This is called the functional equation for ζ (s). The original proof of Riemann for these results is somewhat different from what is stated in the above. In the following, we shall prove similar results for Hecke’s L-functions L(s; χ ), generalizing the method used above.

3.2 Theta-Formulae (Analytic Form) Let n ≥ 1. For x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) in Cn , let (x, y) =

n 

1 2

xk yk , ||x|| = (x, x) =

k=1

 n 

 21 |xk |

2

.

k=1

Let A be an n × n, real, positive definite and symmetric matrix, and put A(x, y) = (x A, y) = (x, y ∗A) =

n 

a jk x j yk ,

j,k=1

A(x) := A(x, x), x, y ∈ Cn . Since A is symmetric, A(x, y) = A(y, x); and since A is real and positive definite, there exists an a > 0 such that A(x) ≥ a x 2 , for any x ∈ Rn . For x, y ∈ Cn , let θ (x, y; A) =



exp(−π A(x + m) + 2πi(y, m)),

m

where m = (m 1 , . . . , m n ) ranges over all integral vectors in Zn . Lemma 3.1 The above series converges absolutely and uniformly in x and y when x and y remain in a bounded domain D of Cn . Hence θ (x, y; A) is a holomorphic function of (x1 , . . . , xn , y1 , . . . , yn ) in C2n . Proof Let q(m; x, y) = −π A(x + m) + 2πi(y, m). Since A(x + m) = A(m) + 2 A(x, m) + A(x), A(x, m) = (x A, m), q(m; x, y) = −π A(m) + (u, m) − π A(x),

38

3 L-functions

where u = −2π x A + 2πi y. For the given bounded domain D in Cn , there exists a constant C > 0 such that

u ≤ C, |π A(x)| ≤ C, x, y ∈ D. 2c 2c . Then |m k | < aπ , m k ∈ Z. Hence there exist Let m = (m 1 , . . . , m n ) ∈ Zn , m < aπ n only a finite number of such m’s in Z . Therefore it is sufficient to show that  exp(q(m; x, y))

m ≥ 2C aπ

converges absolutely and uniformly for x and y in D. For such an m, |(u, m)| + |π A(x)| ≤ u

m + C ≤ C m + C ≤

aπ

m 2 + C, 2

so that Re (q(m; x, y)) = −π A(m) + Re ((u, m) − π A(x)) aπ

m 2 + C ≤ −πa m 2 + 2 πa = − m 2 + C. 2 Hence 

| exp(q(m; x, y))| =

2c

m ≥ aπ





πa exp − m 2 , C  = eC 2 m 

πa  ≤ C exp − m||2 2 m∈Zn  

πa  2   2 =C exp − l 2 l∈Z   ∞

πa  n   2 = C 1+2 exp − l 2 l=1   ∞

πa   ≤ C 1 + 2 exp − l 2 l=1

 πa n e− 2 . = C 1 + 2 πa 1 − e− 2 ≤ C

This proves our assertion.

exp(Re (q(m; x, y)))

m



3.2 Theta-Formulae (Analytic Form)

Lemma 3.2 θ (x, 0; A) =

39

√1 θ (0, x; |A|

A−1 ), x ∈ Cn .

Proof We first note that if A is real, positive definite and symmetric, then so is A−1 , and |A| > 0. Hence θ (x, y; A−1 ) is defined, and is a holomorphic function of x and y. Since both sides of the above are holomorphic in x ∈ Cn , it is sufficient to prove the equality for real x ∈ Rn . Let f (x) = θ (x, 0; A) =



exp(−π A(x + m)), x ∈ Rn .

m

Then f (x + m  ) = f (x) for any m  in Zn , namely, f (x) is periodic in each xk with period 1. Since f (x) is of class C ∞ on Rn , it follows from the standard theorem on Fourier series that  a(m) exp(2πi(x, m)), f (x) = m∈Zn

where the right-hand side converges absolutely and uniformly on each bounded domain of Rn , and where a(m) is given by

1

a(m) =



1

···

0

f (x) exp(−2πi(x, m))d x1 · · · d xn .

0

Now, the above integral is equal to 0

=

1



 m

= =

1

··· 0

1

m

  exp −π A(x + m  ) − 2πi(x, m) d x



1

···

0



0 m 1 +1

m 1 m ∞ −∞



···



··· ∞

−∞

  exp −π A(x + m  ) − 2πi(x + m  , m) d x

m n +1

m n

exp (−π A(x) − 2πi(x, m)) d x

exp (−π A(x) − 2πi(x, m)) d x1 · · · d xn .

Since A is positive definite, there exists an n × n, real, positive definite and symmetric matrix B such that A = B 2 . Let y = x B, t = m B −1 . Then A(x) = (x A, x) = (x B 2 , x) = (x B, x B) = (y, y), (x, m) = (y B −1 , m) = (y, m B −1 ) = (y, t), so that A(x) + 2i(x, m) = (y, y) + 2(y, it) = (y + it, y + it) + (t, t), where (t, t) = (m B −1 , m B −1 ) = (m B −2 , m) = (m A−1 , m) = A−1 (m). Since  dy1 · · · dyn = |B|d x1 · · · d xn = |A|d x1 · · · d xn ,

40

3 L-functions

we see that the above integral is equal to   1 exp −π A−1 (m) √ |A|



∞ −∞

···

∞

−∞



 n  2 exp −π (yk + itk ) dy1 · · · dyn . k=1

However, one sees easily that

∞

−∞

Therefore

  exp −π(z + u)2 dz =



∞ −∞

  exp −π z 2 dz = 1.

  1 exp −π A−1 (m) . a(m) = √ |A|

It follows that θ (x, 0; A) = f (x) =

 m

  1 exp −π A−1 (m) + 2πi(x, m) √ |A|

1 = √ θ (0, x; A−1 ). |A| Proposition 3.2 For any x, y in Cn , 1 θ (x, y; A) = √ exp (−2πi(x, y)) θ (−y, x; A−1 ). |A| Proof In Lemma 3.2, we make the substitution x → x − i y A−1 . Then the exponent of the m-term on the left is −π A(x − i y A−1 + m) = −π A(x + m) − 2π A(x + m, −i y A−1 ) − π A(−i y A−1 ). Here A(x + m, −i y A−1 ) = (x + m, −i y) = −i(x + m, y) A(−i y A−1 ) = (−i y A−1 , −i y) = −(y A−1 , y) = −A−1 (y). Hence −π A(x − i y A−1 + m) = −π A(x + m) + 2πi(y, m) + 2πi(x, y) + π A−1 (y). The exponent of of the corresponding term on the right is −π A−1 (m) + 2πi(x − i y A−1 , m) = −π A−1 (m) + 2πi(x, m) + 2π A−1 (y, m).

3.2 Theta-Formulae (Analytic Form)

Since

41

−π A−1 (m) + 2π A−1 (y, m) − π A−1 (y) = −π A−1 (−y + m),

we obtain the formula stated in the proposition. Remark 3.1 Let y = 0 in the above. Then we have the equality in Lemma 3.2. Let also x = 0. Then 1 θ (0, 0; A) = √ θ (0, 0; A−1 ), |A| namely,

 m

1  exp (−π A(m)) = √ exp(−π A−1 (m)). |A| m

Let n = 1 and A = (x), A(m) = xm 2 , x > 0. Then we obtain 

1 1  , exp(−π m 2 x) = √ exp −π m 2 x x m∈Z m∈Z 

namely, the formula θ (x) =

√1 θ ( 1 ) x x

stated in Sect. 3.1.

3.3 Theta-Formulae (Arithmetic Form) Let F be a number field, n = [F : Q], and let r1 , r2 be as before; n = r1 + 2r2 . Let c1 , . . . , cn be positive reals such that c j = c j+r2 , r1 + 1 ≤ j ≤ r1 + r2 . Put C = (δ jk ck ). Let a be a non-zero ideal of F, α = (a1 , . . . , an ) a basis of a over Z, and Mα = (a (k) j ). Put A = Mα C t M α . n (l) a (l) A is an n × n matrix whose ( j, k)-entry is l=1 j cl ak . Using c j = c j+r2 , r 1 + 1 ≤ j ≤ r1 + r2 , we see easily that A is real and symmetric. However, it can be proved also as follows. For any x = (x1 , . . . , xn ) in Cn , let u(x) = x Mα = (u 1 (x), . . . , u n (x)), n  a (k) u k (x) = j x j , k = 1, . . . , n. j=1 (k) (k) (k+r2 ) Since a (k) , r1 + 1 ≤ k ≤ r1 + r2 , we have j = a j , 1 ≤ k ≤ r1 , a j = a j

42

3 L-functions

u k (x) = u k (x), 1 ≤ k ≤ r1 , u k (x) = u k+r2 (x), u k+r2 (x) = u k (x), r1 + 1 ≤ k ≤ r1 + r2 . As ck = ck+r2 , r1 + 1 ≤ k ≤ r1 + r2 , it follows that A(x, y) = x Aty = x Mα C t(y Mα ) = u(x)C t u(y) r1 r 1 +r2    = ck u k (x)u k (y) + ck u k (x)u k+r2 (y) + u k+r2 (x)u k (y) . k=r1 +1

k=1

Hence A(x, y) = A(y, x) for any x, y in Cn , and A is therefore symmetric. Let x = y ∈ Rn in the above. Then A(x) =

n 

ck |u k (x)|2 , x ∈ Rn .

(3.2)

k=1

Hence A(x) ≥ 0. As A is symmetric, we see that A is real. Furthermore, A(x) = 0 implies u k (x) = 0, k = 1, . . . , n, namely, x Mα = 0. Since |Mα | = 0, we have x = 0. Therefore A is positive definite. Thus A is an n × n, real, symmetric and positive definite matrix. We now write down the formula θ (x, 0; A) = √1|A| θ (0, x; A−1 ) for the above A. Let m = (m 1 , . . . , m n ) ∈ Zn , and let a=

n 

ak m k .

k=1

Since α = (a1 , . . . , an ) is a basis of a, a ranges over a when m ranges over Zn . Also  ( j) u j (m) = nk=1 ak m k = a ( j) . Hence u(m) = (a (1) , . . . , a (n) ), u(x + m) = (u 1 + a (1) , . . . , u n + a (n) ), x ∈ Cn , where u j = u j (x). Therefore θ (x, 0; A)   r  r 1 1 +r2   (k) 2 (k) (k+r2 ) = exp −π ck (u k + a ) + 2 ck (u k + a )(u k+r2 + a ) . a∈a

k=1

k=r1 +1

On the other hand, we know that (det Mα )2 = ΔN (a)2 (cf. Sect. 1.1). Hence det A = (det C) |det Mα |2 = c1 · · · cn d N (a)2 , where d = |Δ| = N (d). Let β = (b1 , . . . , bn ) be a basis of F/Q, complementary to α. Then β is a basis of the ideal  a = a−1 d−1

3.3 Theta-Formulae (Arithmetic Form)

43

over Z, and Mβ = tMα−1 . It follows that A−1 = A

−1

−1

= tMα−1 C −1 M α = Mβ C −1t M β .

Let v(x) = x Mβ , x = (x1 , . . . , xn ) ∈ Cn . Then A−1 (x) =

n 

ck−1 |vk (x)|2 , x ∈ Rn ,

k=1

 = nk=1 bk m k ranges just as in (3.2). When m = (m 1 , . . . , m n ) ranges over Zn , b  n ck−1 |b(k) |2 . Furover  a = a−1 d−1 . Since v(m) = (b(1) , . . . , b(n) ), A−1 (m) n = k=1 t −1 (k) thermore, (x, m) = (x Mα , m Mα ) = (u(x), v(m)) = k=1 u k b . Therefore 

−1

θ (0, x; A ) =

 exp −π

b∈a−1 d−1

n 

ck−1 |b(k) |2

+ 2πi

k=1

n 

 (k)

b uk ,

k=1

and we have   r  r 1 1 +r2   (k) 2 (k) (k+r2 ) exp −π ck (u k + a ) + 2 ck (u k + a )(u k+r2 + a ) a∈a

=

k=1



1 c1 · · · cn d N (a)2



k=r1 +1

exp −π

b∈a−1 d−1

n 

ck−1 |b(k) |2

+ 2πi

k=1

n 

 (k)

b u k . (3.3)

k=1

The equality holds for any u = u(x) = x Mα , x ∈ Cn , and hence also for arbitrary u = (u 1 , . . . , u n ) in Cn . Let e1 , . . . , en be rational integers such that ek ≥ 0, 1 ≤ k ≤ n, and ek = 0 or 1, 1 ≤ k ≤ r1 , ek ek+r2 = 0, r1 + 1 ≤ k ≤ r1 + r2 . Let ek = ek , 1 ≤ k ≤ r1 ,  ek = ek+r2 , ek+r = ek , r1 + 1 ≤ k ≤ r1 + r2 . 2

We differentiate the both sides of (3.3) ek -times in each variable u k , 1 ≤ k ≤ n; we may differentiate them term by term because the both series are absolutely and uniformly convergent in u in a bounded domain of Cn . Hence we have

44

3 L-functions n   (−2π ck (u k + a (k) ))ek exp(−π(· · · )) a∈a k=1

=

n  

1 c1 · · · cn

d N ((a))2

(2πib(k) )ek exp(· · · ),

b∈a−1 d−1 k=1

namely n 

e

ckk

n 



(u k + a (k) )ek exp(−π(· · · ))

a∈a k=1

k=1

=

n  

(−i)e c1 · · · cn d N (a)2

(b(k) )ek exp(· · · ),

b∈a−1 d−1 k=1

  where e = nk=1 ek = nk=1 ek . Let r be any element of F. Put u k = r (k) in the above. Then we obtain the following result. Proposition 3.3 Let a be a non-zero ideal of F, and let r be an element of F. Then n 

e ckk

k=1

=

n 

 ek

(r (k) + a (k) ) exp −π

a∈a k=1

(−i)

e

c1 · · · cn d N (a)2

n 

 ck |r (k) + a (k) |2

k=1 n   b∈a−1 d−1

(k) ek



(b ) exp −π

k=1

n 

 ck−1 |b(k) |2

+ 2πi T (r b).

k=1

  Here c1 , . . . , cn > 0, c j = c j+r2 for r1 + 1 ≤j ≤ r1 + r 2 , e1 , . . . en and e1 , . . . , en , n n  are integers as stated in the above, and e = k=1 ek = k=1 ek . Furthermore, the series on the both sides of the equality are absolutely convergent.

Now, let χ be a Hecke character of F, f the conductor of χ , f = N (f), and {m 1 , . . . , m n ; s1 , . . . sn } the signature of χ . Then the conjugate χ of χ has the same conductor f and the signature {m 1 , . . . , m n ; −s1 , . . . , −sn } where 1 ≤ k ≤ r1 , m k = m k , m k = m k+r2 , m k+r2 = m k ,

r1 + 1 ≤ k ≤ r1 + r2 .

Let α = (a P ) be any adèle of F; α ∈ R, and let tk = σk (a P∞,k ), 1 ≤ k ≤ r , tk+r2 = t k , r1 + 1 ≤ k ≤ r = r1 + r2 . Let   n n  π  2 m k tk exp − √ |tk | , α ∈ R. g(α; χ ) = χ f (α) n d f k=1 k=1 For any idèle α of F (α ∈ J ), we then define

3.3 Theta-Formulae (Arithmetic Form)

45



θ (α; χ ) =

g(αa; χ ), a = ι (α).

(3.4)

a∈a−1

When a runs over a−1 f, and r over a set of representatives of a−1 modulo a−1 f, then r + a obviously runs over all elements of a−1 (once over each element). Hence θ (α; χ ) =

  r

g(α(r + a); χ ).

a∈a−1 f

Since αa, α(r + a) ∈ R  , αa = α(a + r ) mod Rf , we have χf (αr ) = χf (α(r + a)). Hence g(α(a + r )) = χ f (αr )



m tk k





(r

(k)

+a

(k) m k

)

 n π  2 (k) (a) 2 exp − √ |tk | |r + a | , n d f k=1

and consequently, θ (α; χ ) =



m k

tk



 

χ f (αr )



(r (k) + a (k) )m k exp(· · · ).

a∈a−1 f

r

If we replace a by a−1 f, and put |tk |2 , ek = m k , ek = m k , ck = √ n df

1 ≤ k ≤ n,

 in Proposition 3.3, we see that the series a∈a−1 f is absolutely convergent. Since r ranges over a finite set, it follows that the right-hand side of (3.4) is also absolutely convergent, and hence θ (α; χ ) is well-defined. Furthermore, the same proposition shows that θ (α; χ ) =



m k

tk

 r



×

b∈ad−1 f−1



−1

(−i) M  c1 · · · cn d N (a−1 f)2   n   −1 (k) 2 (k) m k σk |b | + 2πi T (r b) (b ) exp −π

χ f (αr )

m

ck k

k=1

 2  m n √ n   √ d f b(k) k π  n d f b(k) =h exp − √ n tk tk d f k=1 −1 −1 b∈ad f  × χ f (αr )ω0 (r b), r

46

3 L-functions

where (−i) M

h= c1 · · · cn d N (a−1 f)2



m tk k



 m  −1 ck k

 tk m k , √ n df

M=

n 

mk ,

k=1

ω0 (r b) = exp(2πi T (r b)). However, by Proposition 2.5, we know 

χ f (αr )ω0 (αr )ω0 (r b) = χf (α −1 δϕb)C(χ ; δ, ϕ),

r

for any b in ad−1 f−1 . Hence θ (α; χ ) = C(χ ; δ, ϕ) h



g(α −1 δϕb; χ ), ι (α −1 δϕ) = a−1 df,

b∈ad−1 f−1

= C(χ ; δ, ϕ)hθ (α −1 δϕ; χ ).  ι (α0 ), we have V (α0 )  = N (a)−1 for α = α0 α∞ . Also nk=1 |tk | Now, as a = ι (α) =   1 1 +r2 = rk=1 |σk (a P∞,k )| · rk=r |σk (a P∞,k )|2 = rk=1 v P∞,k (a P∞,k ) = V (α∞ ). Hence 1 +1 c1 · · · cn d N (a−1 f)2 =

n 

|tk |2 ·

k=1

1 · d · N (a)−2 f −2 df

= V (α∞ )2 · V (α0 )2 · f = V (α)2 f. On the other hand, since 

m tk k





 m  −1 ck k

tkm k is the complex conjugate of



m

tk k , we have

  tk m k  |tk |2 m k  m  −1 = = 1. ck k √ √ n n df df

Hence h=

(−i) M √ , V (α) f

and the following result is proved. Proposition 3.4 Let α be any idèle of F, and let δ and ϕ be the fixed idèle as defined in Sect. 2.3. Then θ (α; χ ) = W  (χ ; δ, ϕ)V (α)−1 θ (α −1 δ0 ϕ; χ ),

3.3 Theta-Formulae (Arithmetic Form)

where

47

 (−i) M mk , W  (χ ; δ, ϕ) = √ C(χ ; δ, ϕ), M = f k=1 n

depends only upon χ , δ, and ϕ, and is independent of α. By Proposition 2.4, Sect. 2.3, we have  |C(χ ; δ, ϕ)| = f , C(χ ; δ, ϕ) = χf (−1)C(χ ; δ, ϕ). Hence

and

|W  (χ ; δ, ϕ)| = 1, iM W  (χ ; δ, ϕ) = √ χf (−1)C(χ ; δ, ϕ) = (−1) M χf (−1)W  (χ ; δ, ϕ). f

However, since χf (ξ ) = χ (ξ ) for ξ in U0 , and χ (F ∗ ) = 1, we see that χf (−1) = χf ((−1)0 ) = χ ((−1)0 )  n −1  −1 mk = χ ((−1)∞ ) = (−1) = ((−1) M )−1 . k=1

Therefore

W  (χ ; δ, ϕ) = W  (χ; δ, ϕ).

3.4 The Function f (α, s; x) Let F be a number field, n = [F : Q], and let χ be a Hecke character of F with conductor f and signature {m 1 , . . . , m n ; s1 , . . . , sn }. The complex conjugate χ of χ is then also a Hecke character of F, and its conductor is f and its signature is {m 1 , . . . , m n ; −s1 , . . . , −sn }, when m k = m k , 1 ≤ k ≤ r1 , m k = m k+r2 , m k+r2 = m k , r1 + 1 ≤ k ≤ r1 + r2 . For any idèle α of F (α ∈ J ), and for any complex number s, define  χ (α)g(α; χ )V (α)s , if a = ι (α) ⊂ o, f (α, s; χ ) = 0, otherwise. Here V (α) is the volume of α, and g(α; χ ) is the function defined in Sect. 3.3. Namely, let α = (a P ), tk = σk (a P∞,k ), 1 ≤ k ≤ r , tk+r2 = t k , r1 + 1 ≤ k ≤ r . Then

48

3 L-functions

g(α; χ ) = χ f (α)

n 



m tk k

k=1

 n π  2 exp − √ |tk | . n d f k=1

Let α = α0 α∞ , α0 ∈ J0 , α∞ ∈ J∞ . Then χ (α) = χ (α0 )χ (α∞ ), χf (α) = χf (α0 ), V (α) = V (α0 )V (α∞ ), and a = ι (α) = ι (α0 ). Hence f (α, s; χ ) = f 0 (α0 , s; χ ) f ∞ (α∞ , s; χ ), where  χ (α0 )χ f (α0 )V (α0 )s , if a = ι (α0 ) ⊂ o, f 0 (α, s; χ ) = 0, otherwise, 

 m π 2 k |tk | V (α∞ )s . tk exp − √ f ∞ (α∞ , s; χ ) = χ (α∞ ) n df We first consider f 0 (α0 ) = f 0 (α0 , s; χ ). Suppose that a = ι (α 0 ) ⊂ o. Let P1 , . . . , Pg be all the finite primes such that ν P j (f) > 0. Since a = pνPP (a P ) , a is prime to f if and only if ν P j (a P j ) = 0 for j = 1, . . . , g, namely, if and only if (α0 )f = (a P1 , . . . , a Pg , 1 . . . , 1, . . .) is in U0 . In such a case, χf (α0 ) = χ ((α0 )f ) so that χ (α0 )χ f (α0 ) = χ (α0 /(α0 )f ). However, α0 /(α0 )f = (1, . . . , 1, a P , . . .) is in J0 (f), and ι ((α0 )/(α0 )f ) = ι (α0 )ι ((α0 )f )−1 = ι (α0 ) = a. Hence, by the definition of χ : Jf → C∗1 , χ (α0 /(α0 )f ) = χ (a), namely, χ (α0 )χ f (α0 ) = χ (a). On the other hand, if a is not prime to f, then χf (α0 ) = 0 and χ (a) = 0 by the definition of χf (α0 ) and χ (a). Hence we still have χ (α0 )χ f (α0 ) = χ (a). Since V (α0 ) = N (a)−1 , we obtain  χ (a)N (a)−s , if a = ι (α0 ) ⊂ o, f 0 (α0 ) = 0 otherwise. The homomorphism ι : J0 → I induces an isomorphism J0 /U0 ∼ = I. The above equalities shows that f 0 (α0 ) depends only upon the coset α 0 = α0 U0 in J0 /U0 ; f 0 (α0 ) = f 0 (α 0 ). Since U0 is open in J0 , we see in particular that f 0 (α0 , s; χ ) is a continuous function of (α0 , s) in J0 × C. Let μ J0 , μU0 , and μ J0 /U0 be the Haar measures of the respective groups fixed in Sect. 2.4. Since μ J0 = μ J0 /U0 ∗ μU0 , we have





| f 0 (α0 )|dμ J0 (α0 ) = J0

J0 /U0

 | f 0 (α0 ξ )|dμU0 (ξ ) dμ J0 /U0 (α 0 ).

U0

However, f 0 (α0 ξ ) = f 0 (α0 ) for any ξ in U0 , and μU0 (U0 ) = 1. Hence the inner integral on the right is | f 0 (α0 )|. As μ J0 /U0 is the point measure on the discrete group J0 /U0 , we see that

3.4 The Function f (α, s; x)

49



| f 0 (α0 , s; χ )|dμ J0 (α0 ) = J0

=

J0 /U0

| f 0 (α0 )|dμ J0 /U0 (α 0 )



| f 0 (α 0 )|

α 0 ∈J0 /U0

=



|χ (a)N (a)−s |.

a⊂o

 It follows from Sect. 3.1 that J0 | f 0 (α0 , s; χ )|dμ J0 (α0 ) < +∞ if Re (s) > 1. We also see from the above that in such a case,  f 0 (α0 , s; χ )dμ J0 (α0 ) = χ (a)N (a)−s J0

a⊂o

= L(s; χ ). We shall next consider f ∞ (α∞ ) = f ∞ (α∞ , s; χ ). By the definition of the signature, we have  n  tk m k χ (α∞ ) = |tk |isk . |t | k k=1 We also know that V (α∞ ) =

n k=1

|tk |. Hence

 

n  tk m k π m 2 |tk |s |t |tk |isk tk k exp − √ | k n |t | d f k k=1 

n  π s+m k +isk 2 |tk | = |tk | exp − √ n df k=1 

r  π  2 |t |tk |s+m k +isk exp − √ | = k n df k=1 

r  2π 2s+m k +m k +2isk 2 |tk | . |tk | exp − √ × n df k=r +1

f ∞ (α∞ ) =

1

This shows in particular that f ∞ (α∞ , s; χ ) is a continuous function of (α∞ , s) in J∞ × C. Hence f (α) = f 0 (α0 ) f ∞ (α∞ ) is also continuous on J = J0 × J∞ . Let μ J∞ be the Haar measure of J∞ fixed in Sect. 2.4. It follows from the definition of μ J∞ that J∞

f ∞ (α∞ )dμ J∞ (α∞ ) = I1 I2 · · · Ir ,

50

3 L-functions

where 

π  2 dμ(t), 1 ≤ k ≤ r1 , |t| |t|s+m k +isk exp − √ n df R∗

 2π  2 |t| |t|2s+m k +m k +2isk exp − √ = dμ(t), r1 + 1 ≤ k ≤ r, n df C∗

Ik =

μ being the standard measures on R∗ and C∗ respectively. . Hence Let 1 ≤ k ≤ r1 . Then dμ(t) = dt |t| Ik =



0 −∞

+∞

+ 0



∞

=2

t

s+m k +isk

0



π 2 dt t exp − √ n t df

 1 (s+m k +isk ) ∞

√ n df 2 du 1  = − u 2 (s+m k +isk ) exp(−u) , π u 0

with u = − √n πd f t 2 . Since Re (s + m k + isk ) = Re (s) + m k ≥ Re (s), we see that if Re (s) > 0, then the last integral is absolutely convergent, and  1 (s+m k +isk )

√  n df 2 1 (s + m k + isk ) . Ik = Γ π 2 If r1 + 1 ≤ k ≤ r , then dμ(t) = 2drr dθ , where t = r eiθ , r = |t|. Hence by a similar computation, we see that if Re (s) > 0, then the integral for Ik converges absolutely, and  k +is s+ mk +m 

√

k n 2 m k + m k df + isk . Γ s+ Ik = 2π 2π 2 Thus we see that J∞

| f ∞ (α∞ , s; χ )|dμ J∞ (α∞ ) < +∞

for Re (s) > 0, and f ∞ (α∞ , s; χ )dμ J∞ (α∞ ) = I1 · · · Ir J∞

= (2π )r2

 ek r √ n  df 2 k=1

ek π

 m +m  s+ k 2 k +isk

Γ



m k + m k ek , s+ + isk − 2 2

where ek = 1 or 2 according as k ≤ r1 or k > r1 . We may also write it in the form

3.4 The Function f (α, s; x)

51

J∞

f ∞ (α∞ , s; χ )dμ J∞ (α∞ ) = A(s; χ )γ (s; χ ),

where A(s; χ ) = (2π )r2

 ek r √ n  df 2 k=1

γ (s; χ ) =

r 

Γ

k=1

ek 2

 m +m  s+ k 2 k +isk

ek π

,



m k + m k . s+ + isk 2

For example, if χ ≡ 1, then m k = m k = sk = 0, 1 ≤ k ≤ r , so that  √ A(s; 1) = (2π )r2 γ (s; 1) = Γ

s r1 2

d n r 2 2 π2

s ,

Γ (s)r2 .

Now, since f (α, s; χ ) = f 0 (α0 , s; χ ) f ∞ (α∞ , s; χ ), J = J0 × J∞ , μ J = μ J0 × μ J∞ , we immediately obtain from the above the following result. Proposition 3.5 The function f (α, s; χ ) is continuous on J × C. For Re (s) > 1, it is integrable on J : | f (α, s; χ )|dμ J (α) < +∞, Re (s) > 1. J



Let

f (α, s; χ )dμ J (α), Re (s) > 1.

ξ(s; χ ) = J

Then ξ(s; χ ) = A(s; χ )γ (s; χ )L(s; χ ), Re (s) > 1, where A(s; χ ) and γ (s; χ ) are the functions defined in the above. Let O = {α | α ∈ J, V (α) > 1}. O is an open set in J . Proposition 3.6 For any complex s, f (α, s; χ ) is integrable on O, and f (α, s; χ )dμ(α)

η(s; χ ) = O

defines a holomorphic function of s on the entire s-plane.

52

3 L-functions

Proof Given any s ∈ C, find c > 1 such that Re (s) ≤ c. As  | f (α, s; χ )| =

|χ (α)g(α; χ )V (α)s | = |g(α; χ )|V (α)Re (s) , or 0,

we see that | f (α, s; χ )| ≤ | f (α, c; χ )| for α in O; V (α) > 1. Hence | f (α, s; χ )|dμ J (α) ≤ | f (α, c; χ )|dμ J (α) O O | f (α, c; χ )|dμ J (α) < +∞, ≤ J

by Proposition 3.5. This proves the first half. At the same time, it was shown that | f (α, s; χ )|dμ J (α) ≤ K c < +∞

|η(s; χ )| ≤ O

whenever Re (s) ≤ c. Let C be an arbitrary circle on the s-plane, and let c > 1 be chosen so that Re (s) ≤ c for s ∈ C. For fixed α ∈ O, f (α, s; χ ) is of course holomorphic in s. Hence, for any s inside of the circle C, we have f (α, s; χ ) = Therefore η(s; χ ) = However,



1 2πi

1 2πi

O

C

C

f (α, z; χ ) dz. z−s

 f (α, z; χ ) dz dμ J (α). z−s

f (α, z; χ ) Kc dμ J (α)dz ≤ dz < +∞. z−s |z − s| O×C C

Hence, by Fubini’s Theorem,

 1 1 f (α, z; χ )dμ J (α) dz 2πi C z − s O η(z; χ ) 1 dz. = 2πi C z − s

η(s; χ ) =

3.4 The Function f (α, s; x)

53

This holds for any s inside of C. As |η(z; χ )| ≤ K c on C, we see that η(s; χ ) is holomorphic in s inside of C, and hence it is holomorphic everywhere on the splane. Let μ F ∗ denote as before the point measure of the discrete subgroup F ∗ of J , and let μ J = μ J /μ F ∗ , J = J/F ∗ . Since V (αa) = V (α), α ∈ J , a ∈ F ∗ , O is a union of cosets mod F ∗ . Hence  · · · dμ J = · · · dμ F ∗ dμ J , O

O

F∗

where O denotes the image of O under J → J = J/F ∗ . In particular, η(s; χ ) =

F∗

O

 f (αa, s; χ )dμ F ∗ (a) dμ J (α).

Since μ F ∗ is the point measure, F∗

f (αa, s; χ )dμ F ∗ (a) =



f (αa, s; χ )

a∈F ∗

=



χ (αa)g(αa; χ )V (αa)s

−1

a∈a a=0

= χ (α)V (α)s



g(αa; χ )

a∈a−1 a=0

= χ (α)V (α)s (θ (α; χ ) − εχ ),  where θ (α; χ ) = a∈a−1 g(αa; χ ) as in Sect. 3.3, and εχ = g(0; χ ). Hence we have the following formula: χ (α)(θ (α; χ ) − εχ )V (α)s dμ J (α).

η(s; χ ) = O

Note that the functions χ (α), V (α), and θ (α; χ ) depend only upon the coset α = α F ∗ of α mod F ∗ so that they may be also denoted by χ (α), V (α), and θ (α; χ ). Note also that since 

n  π m 2 , |t tk k exp − √ | g(α; χ ) = χ f (α) k n df k=1 

we have εχ = g(0; χ ) =

1, if f = o and m k = 0, 1 ≤ k ≤ n, 0, otherwise.

54

3 L-functions

3.5 Fundamental Theorems Now, we have the disjoint decomposition of J ; J = O ∪ O −1 ∪ J1 , so that





= J

+

O −1

O

+

. J1

However, since J = J1 × T , J1 = J1 × {1}, μ  J = μ J1 × μT , and since the point 1 has measure 0 in T , we see that μ J (J1 ) = 0, J1 = 0. Hence, in particular, ξ(s; χ ) =

f (α, s; χ )dμ J (α)



J

f (α, s; χ )dμ J (α) +

= O

O −1

f (α, s; χ )dμ J (α),

for any s with Re (s) > 1. The integral over O is equal to η(s; χ ). For the integral over O −1 , we obtain f (α, s; χ )dμ J (α) = −1 χ (α)(θ (α; χ ) − εχ )V (α)s dμ J (α), O −1

O

in the same way as we have done in Sect. 3.4. However, θ (α; χ ) = W  (χ ; δ, ϕ)V (α)−1 θ (α −1 δϕ; χ ) by Proposition 3.4, Sect. 3.3. Hence

O −1

f (α, s; χ )dμ J (α) = O

−1

χ (α)(W  V (α)−1 θ (α −1 δϕ; χ ) − εχ )V (α)s dμ J (α).

It follows from the last remark in Sect. 3.4 that εχ = εχ = 1 or 0 and that if εχ = 1, then f = o, m k = 0, 1 ≤ k ≤ n, so that 

(−i) W (χ ; δ, ϕ) = − √ 

f

mk

C(χ ; δ, ϕ) = 1.

Hence we may write the integrand of the last integral in the form W  χ (α)(θ (α −1 δϕ; χ ) − εχ )V (α)s−1 + εχ χ (α)(V (α)s−1 − V (α)s ). Replace α by α −1 δϕ in the first term of the above. Since δ, ϕ ∈ J1 , V (δϕ) = 1, we then obtain W  χ (δϕ)χ (α)(θ (α; χ ) − εχ )V (α)1−s .

3.5 Fundamental Theorems

55

−1

As O −1 → O (O → O) under α → α −1 δϕ (α → α −1 δϕ), we see that W  χ (α)(θ (α −1 δϕ; χ ) − εχ )V (α)s−1 dμ J (α) −1 O χ(α)(θ (α; χ ) − εχ )V (α)1−s dμ J (α) =W O

= W η(1 − s; χ ), with

W = W (χ ) = χ (δϕ)W  (χ ; δϕ).

Problem 3.1 Prove that W (χ ) depends only upon χ , and is independent of the choice of δ and ϕ such that δ, ϕ ∈ J1 ∩ (J0 × T ), ι (δ) = d, ι (ϕ) = f. It follows that ξ(s; χ ) = + O

O −1



= η(s; χ ) + W η(1 − s; χ ) +

O −1

  εχ χ (α) V (α)s−1 − V (α)s dμ J (α),

for any s with Re (s) > 1. The computation also shows that the last integral converges for Re (s) > 1. Now, let α = α 1 · τ x , α 1 ∈ J 1 , τ x ∈ T , x ∈ R+ , for α in J = J 1 × T . Then V (α) = V (τ x ) = x, and we have O

−1

= {α | α ∈ J , V (α) < 1} = J 1 × {τ x | 0 < x < 1}.

Since dμT (τ x ) → O

dx x

under T ∼ = R+ , we see that

  εχ χ (α) V (α)s−1 − V (α)s dμ J (α) 1 dx εχ χ (α 1 )dμ J 1 (α 1 ) χ (τ x )(x s−1 − x s ) , = x 0 J1

−1

(χ (τ x ) = 1).

Suppose first that χ ≡ 1. Then εχ = 1, and the above integrals become

56

3 L-functions



J1

dμ J 1 (α 1 )

1

dx x  s 1

(x s−1 − x s )

0

 s−1 x x − = μ J 1 (J 1 ) · s−1 s 

1 1 − = μ J 1 (J 1 ) s−1 s 1 μ (J 1 ). = s(s − 1) J 1

(∵ Re (s) > 1)

0

Since this integral must have a finite value, we see that μ J 1 (J 1 ) < +∞. Suppose next that χ ≡ 1. As χ is a Hecke character, χ (F ∗ × T ) = 1. Hence χ must induce a non-trivial character on J 1 = J1 /F ∗ . Let β 1 be an element of J 1 such that χ (β 1 ) = 1. Since μ J 1 (J 1 ) < +∞, the bounded continuous function χ (α 1 ), α 1 ∈ J 1 , is integrable on J 1 , and χ (α 1 )dμ J 1 (α 1 ) = χ (α 1 β 1 )dμ J 1 (α 1 ) J1 J1 = χ (β 1 ) χ (α 1 )dμ J 1 (α 1 ). J1

Then it follows from χ (β 1 ) = 1 that χ (α 1 )dμ J 1 (α 1 ) = 0, J1





and that J1

εχ χ (α 1 )dμ J 1 (α 1 ) = εχ

Thus we see that

J1

χ (α 1 )dμ J 1 (α 1 ) = 0.

ξ(s; χ ) = η(s; χ ) + W (χ )η(1 − s; χ ) +

εχ v , Re (s) > 1. s(s − 1)

(3.5)

where v = μ J 1 (J 1 ) < +∞ and εχ = 1 or 0 according as χ ≡ 1 or χ ≡ 1. By Proposition 3.6, the right-hand side of the above is a meromorphic function of s on the entire s-plane; if χ ≡ 1, it has only simple poles at s = 0 and s = 1, and if χ ≡ 1, it is holomorphic everywhere. Hence we see from (3.5) that the function ξ(s; χ ), which was originally defined only for complex s with Re (s) > 1 analytically continued to a meromorphic function of s on the entire s-plane, and satisfies (3.5) for arbitrary s. Now, W (χ ) = χ (δϕ)W  (χ ; δ, ϕ) by the definition. Hence it follows from Proposition 2.4, that

3.5 Fundamental Theorems

57

|W (χ )| = |χ (δϕ)||W  (χ ; δ, ϕ)| = 1, W (χ ) = χ (δϕ)W  (χ; δ, ϕ) = W (χ ), so that W (χ )W (χ ) = 1. On the other hand, if χ ≡ 1, then W (1) = 1. Hence we obtain from (3.5) that

ξ(s; χ ) = W (χ ) W (χ )η(s; χ ) + η(1 − s; χ ) +

εχ v s(s − 1)



= W (χ )ξ(1 − s; χ ), namely, ξ(s; χ ) = W (χ )ξ(1 − s; χ ). By Proposition 3.5, we have L(s; χ ) = A(s; χ )−1 γ (s; χ )−1 ξ(s; χ ).

(3.6)

for Re (s) > 1. Here A(s; χ )−1 is a function of the form eas+b , where a, b = constant, and γ (s; χ )−1 is a product of functions of the form Γ (cs + d)−1 . Since Γ (s)−1 is holomorphic on the entire s-plane, A(s; χ )−1 γ (c; χ )−1 is also holomorphic for arbitrary s. It then follows from the above that the L-function L(s; χ ), which was originally defined for s with Re (s) > 1, is a meromorphic function of s on the entire s-plane, satisfying (3.6) for arbitrary s. If χ ≡ 1, then ξ(s; χ ) is an entire function of s. Hence L(s; χ ) is also an entire function of s. Let χ ≡ 1. ζ F (s) = L(s; 1) is holomorphic at s = 0, 1, and since γ (s; 1)−1 = Γ

s −r1 2

Γ (s)−r2 , r1 + r2 > 0,

has a zero at s = 0, ζ F (s) is still holomorphic at s = 0. At s = 1, ξ(s; 1) has a simple pole with residue v = μ J 1 (J 1 ), and r1

π2 A(1, 1)−1 = √ , γ (1; 1)−1 = Γ d

−r1 r1 1 = π− 2 . 2

Hence ζ F (s) has a simple pole at s = 1 with residue We now summarize our results as follows.

√v . d

Theorem 3.1 The L-function L(s; χ ) for a Hecke character χ of F, which was originally defined for s with Re (s) > 1, is a meromorphic function of s on the entire s-plane. If χ ≡ 1, then L(s; χ ) is holomorphic everywhere (i.e., an entire function of s). If χ ≡ 1, then ζ F (s) = L(s; 1) has a unique simple pole at s = 1 with residue √v , v = μ (J 1 ) < +∞. Let J1 d

58

3 L-functions

ξ(s; χ ) = A(s; χ )γ (s; χ )L(s; χ ), s ∈ C, with A(s; χ ) and γ (s; χ ) in Sect. 3.3. Then ξ(s; χ ) satisfies the functional equation ξ(s; χ ) = W (χ )ξ(1 − s; χ ), s ∈ C, where W (χ ) is a constant, depending only upon χ , such that |W (χ )| = 1, W (χ ) = W (χ ). Now, since m k , m k ≥ 0 and sk are real, γ (s; χ ) =

r 

Γ

k=1

ek 2

 m k + m k s+ + isk 2

does not vanish for Re (s) > 1. Hence ξ(s; χ ) = A(s; χ )γ (s; χ )L(s; χ ) also has no zero for Re (s) > 1. By the functional equation, we then see that ξ(s; χ ) has no zero for Re (s) < 0. Hence the zeros of L(s; χ ) in the domain Re (s) < 0 are obtained from those of γ (s; χ )−1 , and they can be easily determined because Γ (s)−1 has only simple zeros at s = 0, −1, −2, . . .. Thus all “non-trivial" zeros of L(s; χ ) are in the critical strip {s | 0 ≤ Re (s) ≤ 1}. The generalized Riemann hypothesis states that all such zeros are on the straight line Re (s) = 21 . In general, a locally compact group is compact if and only if it has a finite total Haar measure. In the above, we have shown that μ J 1 (J 1 ) < +∞. Hence Theorem 3.2 The group J 1 = J1 /F ∗ is a compact group. We shall next show that the compactness of J 1 implies two fundamental theorems on algebraic number fields, namely, the finiteness of class numbers and Dirichlet’s unit theorem. (A)

(B)

J

F∗ J 

F∗ × T

J

(E × T )U J

E×T

U

E×T

W

1

1

3.5 Fundamental Theorems

59

Let J  = U0 × J∞ and U be as before. Consider the diagrams in the above.  ⊂ J  , we have (F ∗ × T )J  = F ∗ J  , and (F ∗ × T ) ∩ J  = (F ∗ ∩ Since T ⊂ J∞  J ) × T = E × T , where E = F ∗ ∩ J  is the group of units of F. Since U ⊂ J1 and J = J1 × T , we have (E × T ) ∩ U = E ∩ U = W , which is the group of all roots of unity in F. Since J  is open in J , F ∗ J  is an open (hence also closed) subgroup of J . Therefore J/F ∗ J  is discrete. Since U is compact, (E × T )U is closed, and U/W is compact. As J is separable, the two diagrams gives us topological isomorphisms F ∗ J  /F ∗ × T ∼ = J  /E × T, (E × T )U/E × T ∼ = U/W. Hence (E × T )U/E × T is compact. Let λ : J  → Rr , r = r 1 + r 2 , be defined by α = (a P ) → (log |t1 |, . . . , log |tr |), tk = σk (a P∞,k ). λ is a surjective continuous homomorphism with kernel U . Since J  is separable, λ induces a topological isomorphism λ : J  /U → Rr . ). Then H is the hyperplane consisting of all vectors Let H = λ(J1 ∩ J  ) and L = λ(T  (x1 , . . . , xr ), xk ∈ R, satisfying rk=1 ek xk = 0, where ek = 1 or 2 according as 1 ≤ k ≤ r1 or r1 + 1 ≤ k ≤ r , and L is the straight line generated by ω = ( n1 , . . . , n1 ): L = Rω. Therefore Rr = H ⊕ L . Since E = F ∗ ∩ J  ⊂ J1 ∩ J  , λ(E) is contained in H . Hence λ((E × T )U ) = λ(E × T ) = λ(E) ⊕ L , and λ induces a topological isomorphism J  /(E × T )U ∼ = (H ⊕ L)/(λ(E) ⊕ L) = H/λ(E). On the other hand, λ also induces a topological isomorphism E/W ∼ = EU/U ∼ = λ(E). Since E is discrete, we see that λ(E) is a discrete subgroup of H ∼ = Rr −1 .

60

3 L-functions

Now, since J = J1 × T , we have J 1 = J1 /F ∗ ∼ = J/F ∗ × T. Hence compactness of J 1 ⇐⇒ compactness of J/F ∗ × T ⇐⇒ compactness of J/F ∗ J  and F ∗ J  /F ∗ × T ⇐⇒ compactness of J/F ∗ J  and J  /E × T. Since (E × T )U/E × T is compact, compactness of J  /E × T ⇐⇒ compactness of J  /(E × T )U ⇐⇒ compactness of H/λ(E). Thus

compactness of J 1 ⇐⇒ compactness of J/F ∗ J  and H/λ(E).

Now, since J/F ∗ J  is discrete, the compactness of J/F ∗ J  implies that it is a finite group. However, we know from Chap. 2 that J/F ∗ J  is naturally isomorphic to the ideal class group I/H of F. Therefore the class number h = [I : H] = [J : F ∗ J  ] of F is finite. On the other hand, since λ(E) is a discrete subgroup of H ∼ = Rr −1 , the compactness of H/λ(E) implies that λ(E) is an (r − 1)-dimensional lattice in H∼ = Rr −1 . Hence λ(E) is a free abelian group of rank r − 1, and consequently E/W has the same structure. This is Dirichlet’s unit theorem. Although the above proof may not be simpler than the usual proof of these classical results, it is quite interesting because it clearly indicates the remarkable fact that the two main theorems in algebraic number theory, the finiteness of class numbers and Dirichlet’s unit theorem, are consequences of the compactness  of the group J 1 , which in turn is a consequence of the finiteness of the integral J f (α, s; 1)dμ J (α) for Re (s) > 1, namely, essentially a consequence of the convergence of ζ F (s) =  N (a)−s for Re (s) > 1.

3.6 The Residue of ζ F (s) at s = 1 We shall next indicate how to compute v = μ J 1 (J 1 ) explicitly. By Theorem 3.1, this will give us an explicit value of the residue of ζ F (s) at s = 1. In general, let G be a separable (or σ -compact) locally compact abelian group, and let U and V be closed subgroups of G such that U V is also closed in G. Let μG , μU , μV and μU ∩V be arbitrary Haar measures on G, U , V , and U ∩ V respectively.

3.6 The Residue of ζ F (s) at s = 1

61

G

UV

U

V

U ∩V

1 Lemma 3.3 To each pair of groups X , Y , Y ⊂ X , in the diagram on the above, there exists a unique Haar measure μ X/Y such that (i) For X = G, U , V , U ∩ V , μ X/1 coincides with the given μ X , (ii) For any Z ⊂ Y ⊂ X , μ X/Y · μY/Z = μ X/Z , (iii) μU V /U ≈ μV /U ∩V , μU V /V ≈ μU/U ∩V in the obvious manner. Proof Let      μU/U ∩V = μU /μU ∩V , μV /U ∩V = μV /μU ∩V , μU V /U ∩V = μU/U ∩V × μV /U ∩V ,

and let μU V = μU V /U ∩V × μU ∩V . Let μ1 be the point measure of 1: μ1 (1) = 1. With these μG , μU V , μU , μV , μU ∩V , μ1 , put μ X/Y = μ X /μY , for any X , Y , Y ⊂ X . Then (i) is trivially satisfied. Also μ X/Y · μY/Z = (μ X /μY ) ·  (μY /μ Z ) = μ X /μ Z = μ X/Z (cf. Sect. 2.4). It is known that μU V /U ∩V /μU/U ∩V ≈     μV /U ∩V . However, μU/U ∩V = μU/U ∩V , μV /U ∩V = μU/U ∩V , and μU V /U ∩V = μU V /μU ∩V = μU V /U ∩V . Hence μV /U ∩V ≈ μU V /U ∩V /μU/U ∩V = (μU V /μU ∩V )/ (μU /μU ∩V ) = μU V /μU = μU V /U (Sect. 2.4). Similarly, μU/U ∩V ≈ μU V /V . The uniqueness is obvious, because there is no other choice for μU V and μ1 . In Sect. 2.4, we have fixed the Haar measure μG for G = U0 , J0 /U0 , J0 , J∞ , T , J1 , F ∗ , J , J 1 , and T . Let μ J  = μU0 × μ J∞ μ F ∗ ×T = μ F ∗ × μT , μ E×T = μ E × μT ,

62

3 L-functions

where μ E is the point measure of the discrete group E. We apply the lemma to the diagram (A) of Sect. 3.5, and fix the measures of the groups in that diagram by the above lemma. Since J = J1 × T , we have μ J/(F ∗ ×T ) = μ J /μ F ∗ ×T = (μ J1 × μT )/(μ F ∗ × μT ) ≈ (μ J1 /μT ) × (μT /μT ) = μ J1 /μT = μ J 1 . Hence

v = μ J 1 (J 1 ) = μ J/(F ∗ ×μT ) (J/(F ∗ × T )).

Also μ(F ∗ ×T )/(E×T ) = μ F ∗ ×T /μ E×T = (μ F ∗ × μT )/(μ E × μT ) ≈ (μ F ∗ /μ E ) × (μT /μT ) = μ F ∗ /μ E . Since μ F ∗ and μ E are both point measures, so is μ F ∗ /μ E . Hence μ(F ∗ ×T )/(E×T ) , and consequently μ F ∗ J  /J  are also point measures. As μ J/J  = μ J /μ J  = (μ J0 × μ J∞ )/(μU0 × μ J∞ ) ≈ μ J0 /μU0 = μ J0 /U0 , μ J/J  is a point measure. Since μ J/J  = μ J/F ∗ J  · μ F ∗ J  /J  , we see that μ J/F ∗ J  is also a point measure. Hence v = μ J/(F ∗ ×T ) (J/(F ∗ × T )) = μ J/F ∗ J  (J/F ∗ J  ) · μ F ∗ J  /(F ∗ ×T ) (F ∗ J  /(F ∗ × T )) = [J : F ∗ J  ]μ J  /(E×T ) (J  /(E × T )) = hμ J  /(E×T ) (J  /(E × T )).  be the inverse image of (R+ )r under the isomorphism Let J∞

σ : J∞ → (R∗ )r1 × (C∗ )r2 = ({±1} × Cr12 ) × (R+ )r . Then

 , J∞ = U∞ × J∞

 J  = U0 × J∞ = U × J∞ .

We denote by μU∞ and μ J∞ the translate of the standard measures on ({±1}r1 × Cr12 ) and (R+ )r respectively under the above isomorphism. Then μU∞ (U∞ ) = 2r1 (4π )r2 , and

3.6 The Residue of ζ F (s) at s = 1

63

μ J∞ = μU∞ × μ J∞ , because μ J∞ is, by definition, the translate of the standard measure on (R∗ )r1 × (C∗ )r2 under the same isomorphism. Put μU = μU0 × μU∞ . Then μ J  = μU0 × μ J∞ = μU0 × μU∞ × μ J∞ = μU × μ J∞ . Let μW be the point measure of W so that w = μW (W ) = [W : 1]. With these μ J  , μ E×T , μU , and μW , we apply the lemma to the diagram (B) and fix the measures of the groups which appear in that diagram. Then μ J  /(E×T ) = μ J  /μ E×T coincides with the measure of J  /(E × T ) already fixed in (A). Since μU/W (U/W ) = μU (U )/μW (W ) = μU0 (U0 )μU∞ (U∞ )/μW (W ) = 2r1 (2π )r2 /w, we have μ(J  /(E × T )) = μ(J  /(E × T )U )μ((E × T )U/(E × T )) = μ(J  /(E × T )U )μ(U/W ) = 2r1 (4π )r2 w−1 μ(J  /(E × T )U ). Now, the isomorphism J  /U → Rr considered in the proof of Theorem 3.2 is the log

 → (R+ )r → Rr . Since μ J  = μU × composite of the isomorphisms J  /U → J∞ μ J∞ , we have μ J  /U = μ J  /μU ≈ μ J∞ . Also, the log map translates the standard measure of (R+ )r to the standard measure μRr of Rr . It follows that the translate of μ J  /U under the isomorphism J  /U → Rr is the standard measure μRr of Rr . Let

μ J  /(E×T )U ≈ μRr /(λ(E)⊕L) , μ(E×T )U/U ≈ μλ(E)⊕L , under the isomorphisms induced by J  /U → Rr . Since μ J  /U = μ J  /(E×T )U · μ(E×T )U/U , we have μRr = μRr /(λ(E)⊕L) · μλ(E)⊕L . Since μ(E×T )U/U ≈ μ(E×T )/W = (μ E × μT )/(μW × μ1 ) ≈ μ E /μW × μT , we see that μλ(E)⊕L = μλ(E) × μ L , where μλ(E) is the point measure of λ(E), and μ L is the translate of μT under T → L, namely, the measure defined by dμ L (xw) = d x, x ∈ R. Now, let ε1 , . . . , εr −1 be units in E such that ω1 = λ(ε1 ), . . . , ωr −1 = λ(εr −1 ) form a basis of the lattice λ(E) over Z. Then ω1 , . . . , ωr −1 , ωr form a basis of Rr over R. Let dμRr (ξ ) = d x1 · · · d xr

64

3 L-functions

for ξ = x1 ω1 + · · · + xr ωr in Rr . Then μRr is a Haar measure of Rr , and μRr = cμR for some c > 0. We see immediately that (μRr /μλ(E)⊕L )(Rr /(λ(E) ⊕ L)) = 1. Hence μRr /(λ(E)⊕L) (Rr /(λ(E) ⊕ L)) = (μRr /μλ(E)⊕L )(Rr /(λ(E) ⊕ L)) = c · (μRr /μλ(E)⊕L )(Rr /(λ(E) ⊕ L)) = c, and consequently, μ(J  /(E × T )U ) = μRr /(λ(E)⊕L) (Rr /(λ(E) ⊕ L)) = c, μ(J  /(E × T )) = 2r1 (4π )r2 w−1 c, v = h2r1 (4π )r2 w−1 c. Now, let ξ = x1 ω1 + · · · + xr ωr = (y1 , . . . , yr ). Since ωk = (log |εk(1) |, . . . , log |εk(r ) |), 1 ≤ k ≤ r − 1, we have yj =

r −1 

( j)

xk log |εk | + xr ·

k=1

1 , n

j = 1, . . . , r.

As dμRr (ξ ) = dy1 · · · dyr and dμR (ξ ) = d x1 · · · d xr , we see that ⎛ ⎞ log |ε1(1) | · · · log |ε1(r ) | ⎜ ⎟ .. .. ⎜ ⎟ . . c = det ⎜ ⎟ ⎝ log |ε(1) | · · · log |ε(r ) | ⎠ r −1 r −1 1 1 ··· n

n

= 2−r2 R, where

⎛ ⎞ e1 log |ε1(1) | · · · er log |ε1(r ) | ⎜ ⎟ .. .. ⎜ ⎟ . . R = det ⎜ ⎟ . ⎝ e log |ε(1) | · · · e log |ε(r ) | ⎠ 1 r r −1 r −1 e1 er ··· n n

Since R = 2r2 c = 2r2 μ(J  /(E × T )U ), R is independent of the choice of ε1 , . . . , εr −1 (such that ω1 , . . . , ωr −1 form a basis of λ(E)), and it is an invariant of the field F. We call it the regulator of F. If r > 1, then

3.6 The Residue of ζ F (s) at s = 1

65

⎛ e1 log |ε1(1) | ⎜ .. ⎜ . R = det ⎜ ⎝ e log |ε(1) | 1 r −1 e1 n ⎛ e1 log |ε1(1) | ⎜ .. = det ⎝ . e1 log |εr(1) −1 |

⎞ 0 ⎟ 0⎟ ⎟ (r −1) ⎠ er −1 log |εr −1 | 0 er −1 1 n ⎞ er −1 log |ε1(r −1) | ⎟ .. ⎠ , . (r −1) er −1 log |εr −1 |

· · · er −1 log |ε1(r −1) | .. . ··· ··· ··· ···

because |N (ε)| = 1 for any unit ε in E. From the above computation and from Theorem 3.1, we now have the following theorem. Theorem 3.3 The measure μ J 1 (J 1 ) is given by v = h2r1 +r2 π r2 R/w, and the residue of ζ F (s) at s = 1 is equal to 2r1 +r2 π r2 R h. √ w d Here h is the class number of F, R is the regulator of F, w is the number of roots of unity in F, and d is the absolute value of the discriminant of F. Example 3.1 Let F = Q. Then h = 1. Since r = r1 + r2 = 1 + 0 = 1, R = | en1 | = 1. We also see immediately that w = 2 (W = {±1}) and d = 1. Hence the residue of ζ (s) = ζQ (s) at s = 1 is 1. However, in this special case, we can compute the residue also as follows. Let s > 1. Then 1

∞

∞

 1 dx < 0. In Chap. 2, we defined the subgroups J0 (m) and U0 (m) of J by J0 (m) = {α | α = (a P ) ∈ J0 , νp j (ap j − 1) ≥ tp j , 1 ≤ j ≤ g}, U0 (m) = J0 (m) ∩ U0 = J0 (m) ∩ U (ap = a Pp ). Let I(m) be as before the subgroup of I consisting of all ideals in I which are prime to m. Then the map ι : J → I induces a surjective homomorphism J0 (m) → I(m) with kernel U0 (m) so that we obtain an isomorphism J0 (m)/U0 (m) −→ I(m). Now, let χ be a Hecke character of F with conductor f; f is the largest integral ideal of F such that χ (U0 (f)) = 1. It follows from the above that χ defines a character ψ = ψχ of I(f). Namely, given any ideal a of I(f), we define ψ(a) = χ (α), where α is an idèle of J0 (f) such that a = ι (α). For an integral ideal a of F, we extend the definition of ψ by putting ψ(a) = 0, whenever (a, f) = 1. Thus ψ(a) is defined for a which is either prime to f or integral. In Chap. 3, we denoted the above (generalized) character of ideals by χ  (a) or χ (a). However, in the following, we shall denote it by ψ or ψχ to distinguish clearly from the Hecke character χ . By the definition, © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2019 K. Iwasawa, Hecke’s L-functions, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-13-9495-9_4

67

68

4 Some Applications

L(s; χ ) =

 (1 − ψχ (p)N (p)−s )−1 p

=



ψχ (a)N (a)−s ,

(Re (s) > 1)

a⊂o

Let X denote the group of all Hecke characters of F. (Since X is the group of all characters of J/F ∗ × T , it is a locally compact abelian group in Pontryagin topology. However, in the following we shall not be interested in such a topology on X .) Let m be any non-zero integral ideal of F, and let X (m) denote the set of all Hecke characters χ in X such that χ (U0 (m)) = 1, namely, such that m is contained in the conductor f of χ ; m ⊂ f, f|m. X (m) is obviously a subgroup  of X . If m1 ⊂ m2 then U0 (m1 ) ⊂ U0 (m2 ), and X (m2 ) ⊂ X (m1 ). Clearly, X = m X (m). Let χ ∈ X (m) so that m ⊂ f = conductor of χ . Let ψχ be the character of I(f) corresponding to χ . Since I(m) ⊂ I(f), ψχ induces a character ψχm of I(m). Let Ch(I(m)) denote the group of all characters of I(m). Then χ → ψχm obviously defines a homomorphism X (m) −→ Ch(I(m)). Proposition 4.1 The above homomorphism is injective. Proof We first prove the following lemma. Lemma 4.1 The group J0 (m)F ∗ is everywhere dense in J . Proof Let P1 , . . . , Ph consist of all infinite prime spots of F and of all finite spots of F such that ν P (m) > 0. Let α = (a P ) be any idèle in J , and let 0 < ε < 1. By the approximation theorem, there exists an element a in F such that v P (a P − a) < εv P (a P ), for P = P1 , . . . , Ph . Since ε < 1, we know a = 0, so a ∈ F ∗ . Let α  = (a P ) with a P = 1 for P = P1 , . . . , Ph and a P = a −1 a P for P = P1 , . . . , Ph . Let α  = α  a = (a P ). Then α  ∈ J0 (m), α  ∈ J0 (m)F ∗ , a P = a P for P = P1 , . . . , Ph , and v P (1 −  a P a −1 P ) < ε for P = P1 , . . . , Ph . With small ε > 0, α is then very close to α. Hence ∗ J0 (m)F is dense in J . Now, suppose that ψχm = 1 for a character χ in X (m). Then χ (J0 (m)) = m ψχ (I(m)) = 1. By the definition of a Hecke character, χ (F ∗ ) = 1. Hence χ (J0 (m)F ∗ ) = 1. Since χ is continuous, it follows from the above lemma that

χ (J ) = 1, namely, χ ≡ 1.

Remark 4.1 X (m) → Ch(I(m)) is not surjective. Let σ : J∞ → R∗ × · · · × R∗ × C∗ × · · · × C∗ be defined as before, and let 0 = σ −1 (R+ × · · · × R+ × C∗ × · · · C∗ ). J∞

4.1 Hecke Characters and Ideal Characters

69

0 J∞ is the set of all α = (a P ) in J∞ such that σk (a P∞,k ) > 0 for k = 1, . . . , r1 . It is 0 . an open subgroup of index 2r1 in J∞ , and it contains the subgroup T ; T ⊂ J∞ 0 is the connected component of 1 in J (and in J∞ ). Problem 4.1 Prove that J∞ 0 ) = 1. Let Let X  denote the subgroup of all characters χ in X such that χ (J∞ {m 1 , . . . , m n ; s1 , . . . , sn } be the signature of a character χ of X . Then χ belongs to X  if and only if m k = 0 for r1 + 1 ≤ k ≤ n, and sk = 0 for 1 ≤ k ≤ n. For any non-zero integral idealm, we put X  (m) = X  ∩ X (m). Then X  (m2 ) ⊂  X (m1 ) for m1 ⊂ m2 , and X  = m X  (m). We shall next determine the image of X  (m) under X (m) → Ch(I(m)). Let 0 ); F(m) = F ∗ ∩ (J0 (m) × J∞

F(m) is the subgroup of all a in F; a = 0, such that νp (a − 1) ≥ tp = νp (m), for νp (m) > 0, for 1 ≤ k ≤ r1 . σk (a) > 0 Lemma 4.2 There exists a natural isomorphism 0 0 0 ) −→ J0 (m) × J∞ /F(m)(U0 (m) × J∞ ), J/F ∗ (U0 (m) × J∞

and the both groups are finite. 0 0 is open in J∞ , J0 (m) × J∞ Proof Since J0 (m) (U0 (m)) is open in J0 , and since J∞ 0 ∗ 0 (U0 (m) × J∞ ) is open in J = J0 × J∞ . By Lemma 4.1, F (J0 (m) × J∞ ) is every0 0 0 ) = F ∗ (U0 (m) × J∞ )(J0 (m) × J∞ ). where dense in J . Hence J = F ∗ (J0 (m) × J∞ ∗ 0 0 ∗ 0 On the other hand, F (U0 (m) × J∞ ) ∩ (J0 (m) × J∞ ) = (F ∩ (J0 (m) × J∞ )) · 0 0 ) = F(m)(U0 (m) × J∞ ). Hence we have the above isomorphism. (U0 (m) × J∞ 0 ∗ 0 ) is discrete. However, F ∗ × T ⊂ Since U0 (m) × J∞ is open in J , J/F (U0 (m) × J∞ ∗ 0 ∗ F (U0 (m) × J∞ ), and J/F × T is compact (Theorem 3.2). Hence J/F ∗ (U0 (m) × 0 0 ) is also compact. It then follows that J/F ∗ (U0 (m) × J∞ ) is a finite group. J∞

Now, let r(m) = ι (F(m)); 0 )= r(m) is the group of all principal ideals (a) with a in F(m). Since ι (J0 (m) × J∞ 0 0 ι (J0 (m)) = I(m), and since the kernel of J0 (m) × J∞ → I(m) is U0 (m) × J∞ , we 0 /U0 (m) × see that r(m) is a subgroup of I(m), and that the isomorphism J0 (m) × J∞ 0 ∼ J∞ = I(m) induces isomorphisms 0 0 )/U0 (m) × J∞ −→ r(m), F(m)(U0 (m) × J∞ 0 0 J0 (m) × J∞ /F(m)(U0 (m) × J∞ ) −→ I(m)/r(m).

It follows in particular that I(m)/r(m) is a finite group.

70

4 Some Applications

Proposition 4.2 Under the homomorphism in Proposition 4.1, X  (m) is mapped to the group of all characters of I(m) which are trivial on r(m). Hence X  (m) ∼ = Ch(I(m)/r(m)). Proof Let χ be a Hecke character of F. Then χ (F ∗ × T ) = 1. By the definition, 0 0 ) = 1. Since T ⊂ J∞ , we see that χ belongs to X  (m) if and only if χ (U0 (m) × J∞  ∗ 0 X (m) is the group of all characters of J such that χ (F (U0 (m) × J∞ )) = 1. It then follows from the above isomorphisms 0 0 0 ) −→ J0 (m) × J∞ /F(m)(U0 (m) × J∞ ) −→ I(m)/r(m) J/F ∗ (U0 (m) × J∞

that when χ ranges over X  (m), ψχm gives us all characters of I(m) that are trivial on r(m). Proposition 4.3 X  is the subgroup of all elements of finite order in X . Proof Since I(m)/r(m)is a finite group, so are Ch(I(m)/r(m)) and X  (m). Hence every element in X  = X  (m) has a finite order. Suppose conversely that χ has a 0 then also has a finite order. However, as finite order in X . The restriction χ  of χ on J∞ 0 ∼ + + ∗ ∗ 0 has a finite J∞ = R × · · · × R × C × · · · × C , no non-trivial character of J∞  0   order. Hence χ = 1, χ (J∞ ) = 1, and χ belongs to X . Let m1 ⊂ m2 , both non-zero integral ideals of F. Then X  (m2 ) ⊂ X  (m1 ), I(m1 ) ⊂ I(m2 ), and r(m1 ) ⊂ r(m2 ). Hence we have homomorphisms X  (m2 ) → X  (m1 ), I(m1 ) → I(m2 ), I(m1 )/r(m1 ) → I(m2 )/r(m2 ), and Ch(I(m2 )/r(m2 )) → Ch(I(m1 )/r(m1 )). Using Proposition 4.2, we then obtain the following commutative diagram: X  (m2 )

X  (m1 )

Ch(I(m2 )/I(m2 ))

Ch(I(m1 )/I(m1 )).

Since the map in the upper level is injective, so is the map in the lower level. Let χ be a character in X  (m) with conductor f. Then m ⊂ f, and f is the largest / integral ideal = {0} with the property χ ∈ X  (f). Hence f = m if and only if χ ∈ X  (m ) for any m = m, m ⊂ m ⊂ o. A character in Ch(I(m)/r(m)) (namely, a character of I(m) which is trivial on r(m)) is called primitive if it is not in the image of Ch(I(m )/r(m )) → Ch(I(m)/r(m)) for any m = m, m ⊂ m ⊂ o. It follows from the above that a character χ in X  (m) has conductor m if and only if the corresponding ψχm is primitive in Ch(I(m)/r(m)). In such a case, ψχ (a) = ψχm (a) for a prime to f = m, and for integral a with (a, f) = 1. Therefore in order to obtain all L-functions ψχ (a) = 0  L(s; χ ) = (1 − ψχ (p)N (p)−s )−1 for the characters χ in X  with conductor m, it is sufficient to find all primitive characters of the finite group I(m)/r(m).

4.1 Hecke Characters and Ideal Characters

71

0 denote the closure of F ∗ J 0 in J . Since F ∗ × T ⊂ F ∗ J 0 , J/F ∗ J 0 is a Let F ∗ J∞ ∞ ∞ ∞ compact abelian group. By the definition of X  , X  may be considered as the group of 0 . Let A be the maximal abelian extension of F in C, and let all characters of J/F ∗ J∞ G be the Galois group of A/F; G is a compact abelian group in Krull topology. One 0 is topologically of the fundamental theorems in class field theory states that J/F ∗ J∞  isomorphic to G in a natural manner, or equivalently, that X is naturally isomorphic to the group of all characters of the compact abelian group G. It was C. Chevalley who first introduced the groups J , X , X  etc., and simplified (clarified) the classical class field theory which had been based upon the groups I(m)/r(m) and Ch(I(m)/r(m)), m ⊂ o.

4.2 The Existence of Prime Ideals Let χ be a Hecke character of F. Since L(s; χ ) =

 −1 1 − ψχ (p)N (p)−s , Re (s) > 1, p

we obtain log L(s; χ ) = −



 −s log 1 − ψχ (p)N (p )

p ∞  1 = ψχ (p)m N (p)−ms , Re (s) > 1. m p m=1

(4.1)

Here log z denotes the branch of the log function such that log 1 = 0. The double sum in the above is absolutely convergent, and we see that  ∞  1 m −ms L(s; χ ) = exp ψχ (p) N (p) , Re (s) > 1. m p m=1 Lemma 4.3 For any real ε, y, ε > 0,



L(1 + ε; 1)3 L(1 + ε + i y; χ )4 L(1 + ε + 2i y; χ 2 ) ≥ 1. Proof Let A denote the product of the L-functions in the above. Then it follows from (4.1) that  ∞  A = exp a(p, m) , p m=1

72

4 Some Applications

with a(p, m) =

 1  3 + 4ψχ (p)m N (p)−imy + ψχ (p)2m N (p)−2imy N (p)−m(1+ε) . m

= 0, then Re (a(p, m)) = m3 N (p)−m(1+ε) > 0. If ψχ (p) = 0, then

If ψχ (p)

m −imy

ψχ (p) N (p)

= 1. Hence, let ψχ (p)m N (p)−imy = eiθ , θ ∈ R. Then 1 (3 + 4 cos θ + cos 2θ )N (p)−m(1+ε) m 1 = (3 + 4 cos θ + 2 cos2 θ − 1)N (p)−m(1+ε) m 2 = (1 + cos θ )2 N (p)−m(1+ε) m ≥ 0.

Re (a(p, m)) =



It follows that |A| = exp

 p

Re (a(p, m)) ≥ 1.

m

Proposition 4.4 For any real y, L(1 + i y; χ ) = 0 Proof (1) Suppose that χ 2 ≡ 1. Then ψχ (p) = 0 or ±1 for any prime ideal p, and  −1  −1 1 − N (p)−s 1 − ψχ (p)N (p)−s ⎧  −s −1 −s −2s ⎪ + ··· , ⎨1 − N (p)  = 1 + N (p) + N (p) −2 −s −s = 1 − N (p) = 1 + 2N (p) + 3N (p)−2s + · · · , ⎪ −1 ⎩ 1 − N (p)−2s = 1 + N (p)−2s + · · · . Let Z (s) = L(s; 1)L(s; χ ), (s ∈ C),  −1   = 1 − N (p)−s 1 − ψχ (p)N (p)−s , Re (s) > 1, p

=

∞ 

am m −s , Re (s) > 1.

(4.2)

m=1

Then we see from the above that am ∈ Z, am ≥ 0, and that am ≥ 1 if m = N (p)2 for some prime ideal p. Hence, for any prime number p, there exists m such that m is a power of p and am ≥ 1. It follows that

4.2 The Existence of Prime Ideals

73 ∞ 

am = +∞.

m=1

Now, suppose that L(1; χ ) = 0. Since L(s; 1) has a unique simple pole at s = 1, (Theorem 3.1), Z (s) is then holomorphic at any complex s. Hence Z (s) =

∞  Z (l) (2) (s − 2)l , s ∈ C l! l=0

Here Z (l) (2) can be computed from (4.2): Z (l) (s) = Z (l) (2) =

∞  m=1 ∞ 

am (− log m)l m −s , Re (s) > 1, am (− log m)l m −2 .

m=1

Using am ≥ 0, we then have ∞ ∞   1 (−2)l am (− log m)l m −2 l! m=1 l=0 ∞ ∞ l  2 l −2 am (log m) m = l! l=0 m=1 ∞ ∞  1 = (2 log m)l am m −2 l! m=1 l=0

Z (2) =

= =

∞  m=1 ∞ 

e2 log m am m −2 am

m=1

= +∞. This contradicts the fact that Z (s) is holomorphic at s = 0. Hence L(1; χ ) = 0. (2) To prove the proposition, we may now assume that y = 0 if χ 2 ≡ 1. Under such an assumption, L(s; χ 2 ) is always holomorphic at s = 1 + 2i y. Hence lim L(1 + ε + 2i y; χ 2 ) = L(1 + 2i y; χ 2 ), ε > 0.

ε→0

74

4 Some Applications

Since L(s; 1) has a simple pole at s = 1, we also have lim εL(1 + ε; 1) = a = residue of L(s; 1) at s = 1.

ε→0

Suppose now that L(1 + i y; χ ) = 0. Then L(s; χ ) is holomorphic at s = 1 + i y, and lim

ε→0

L(1 + ε + i y; χ ) L(1 + ε + i y; χ ) − L(1 + i y; χ ) = lim ε→0 ε ε  = L (1 + i y; χ ).

It follows that lim L(1 + ε; 1)3 L(1 + ε + i y; χ )4 L(1 + ε + 2i y; χ 2 )   L(1 + ε + i y; χ ) 4 3 L(1 + ε + 2i y; χ 2 )ε = lim (εL(1 + ε; 1)) ε→0 ε = 0. ε→0

However, this contradicts the previous lemma. Therefore L(1 + i y; χ ) = 0. It follows from the proposition that ξ(s; χ ) = 0 on the line Re (s) = 2. By the functional equation, we then have ξ(s; χ ) = 0 also on the line Re (s) = 0. Therefore the zeros of L(s; χ ) at s = 0 on Re (s) = 0 are given by the zeros of −1

γ (s; χ )

=

r  k=1

 Γ

ek 2

−1  m k + m k s+ + isk 2

on the same line; at s = 0, we have to take into account the fact that ξ(s; χ ) has a m +m  simple pole at s = 0. If Re (s) = 0, then Re ( e2k (s + k 2 k + isk )) = e4k (m k + m k ). m +m 

Hence Γ ( e2k (s + k 2 k + isk ))−1 has zero on Re (s) = 0 only when m k = m k = 0, and in such a case, it has a unique simple zero at s = −isk . Thus the zeros of L(s; χ ) on Re (s) = 0 are explicitly determined. For example, if χ ≡ 1, then m k = m k = sk = 0, 1 ≤ k ≤ r , and γ (s; 1)−1 has a zero of order r at s = 0. Hence ζ F (s) has a zero of order r − 1 (r = r1 + r2 ) at s = 0, but ζ F (s) = 0 for s = 0 with Re (s) = 0. Let m be any non-zero integral ideal of F. Let χ be a character in X  (m) with conductor f, ψ = ψχ the corresponding character of I(f) with ψ(a) = 0 for integral a, not prime to f, and ψχm the restriction of ψχ on I(m). Let Re (s) ≥ a > 21 . Then

4.2 The Existence of Prime Ideals

75

 ∞

∞ 

1

ψ(p)m N (p)−ms ≤ N (p)−am

m

p m=2

p m=2



N (p)−2a 1 − N (p)−a p  1 ≤ N (p)−2a < +∞, −a 1−2 p

=

because



N (a)−2a < +∞ for 2a > 1. Hence

a⊂o

∞  1 ψ(p)m N (p)−ms is a holom p m=2

morphic function of s for Re (s) > 21 , and we obtain from (4.1) that log L(s; χ ) ∼

 p

ψχm (p)N (p)−s .

Here and in the following, we shall use the notation ∼ to indicate the fact that the difference of the both sides of ∼ is holomorphic at s = 1. If p is in I(m), namely, if p | m, then ψχ (p) = ψχm (p). However, there exist only a finite number of p’s such that p|m. Hence it follows from the above that 

log L(s; χ ) ∼

ψχm (p)N (p)−s .

p∈I(m)

Since χ is a character of X  (m), we know from Sect. 4.1 that ψχm (r(m)) = 1 so that ψχm may be considered as a character of I(m)/r(m). Hence, for any c in I(m)/r(m), ψχm (c) is defined. In particular, ψχm (cp ) = ψχm (p) if cp denote the coset of p in I(m)/r(m), and we have 

log L(s; χ ) ∼

ψχm (cp )N (p)−s .

(4.3)

p∈I(m)

Now, when χ ranges over all characters in X  (m), ψχm ranges over all characters of I(m)/r(m) (Proposition 4.2). Therefore it follows from (4.3) that 

m

ψ χ (c) log L(s; χ ) ∼

χ∈X  (m)

=

  χ

p∈I(m)

 

p∈I(m) χ

=

m

ψ χ (c)ψχm (cp )N p−s



ψχ ψχm (c−1 cp )N (p)−s , 

ψ(c−1 cp )N (p)−s ,

p∈I(m) ψ∈Ch(I(m)/r(m))

for any c in I(m)/r(m). Since I(m)/r(m) is a finite group, we know that

76

4 Some Applications





−1

ψ(c cp ) =

ψ∈Ch(I(m)/r(m))

Hence



N = [I(m) : r(m)], 0,

m

ψ χ (c) log L(s; χ ) ∼ N

χ∈X  (m)



if c = cp , if c = cp .

N (p)−s .

p∈c

If χ = 1, then L(s; χ ) is holomorphic at s = 1 (Theorem 3.1), and L(1; χ ) = 0 by the above proposition. Hence log L(s; χ ) ∼ 0. On the other hand, if χ ≡ 1, then L(s; 1) = ζ F (s) has a simple pole at s = 1 (Theorem 3.1) : L(s; 1) = (s − 1)−1 (a0 + a1 (s − 1) + · · · ), a0 = 0. It follows that log L(s; 1) ∼ log

1 . s−1

Since ψχm (c) = 1 for χ ≡ 1, we obtain from the above the following theorem. Theorem 4.1 Let m be any integral ideal of F. Let N be the order of I(m)/r(m), and let c be any coset of I(m)/r(m). Then  p∈c

N (p)−s ∼

1 1 log . N s−1

In particular, each coset of I(m)/r(m) (i.e., each ideal class of the ideal class group I(m)/r(m)) contains infinitely many prime ideals in it. Proof The first half follows  immediately from the above. Let s → 1, s > 1. Then 1 log s−1 → +∞. Hence N (p)−s → +∞ also. Therefore there exist infinitely many p’s in c.

p∈c

Note that for the proof of Theorem 4.1, it is sufficient to know that L(1; χ ) = 0. Let m = o. Then I(o) = I, r(o) ⊂ H so that I/H is a factor group of I(o)/r(o). Hence Corollary 4.1 Each ideal class (in the ordinary sense) of F contains infinitely many prime ideals. We shall see in the next section that for F = Q, the above theorem gives us the classical theorem of Dirichlet on the prime numbers in an arithmetic progression. We also note that the theorem can be refined as follows. Let π(x; c) denote the number of prime ideals p in the coset c with c ∈ I(m)/r(m) such that N (p) ≤ x (x ≥ 2). Then  √ log x 1 x du + O(xe−α n ), π(x; c) = N 2 log u

4.2 The Existence of Prime Ideals

77

which a constant α > 0. Let π(x) denote the number of prime ideals p of F such that N (p) ≤ x (x ≥ 2). Then we obtain from the above that 

x

π(x) = 0

√ log x du + O(xe−α n ). log u

From this, we can deduce that lim

x→∞

π(x) x log x

= 1.

4.3 Dirichlet’s L-functions In this section, we consider the case F = Q. We then have n = 1, r1 = 1, r2 = 0, 0 = σ −1 (R+ ) = T . Hence X = X  . Every ideal of Q is and σ = σ1 : J∞ → R∗ , J∞ principal; given any a in I, there exists a unique element a ∈ Q, a > 0, such that a = (a). The map a = (a) → a defines an isomorphism I −→ Q+ = the multiplicative group of positive rationals. The integral ideals in I correspond to positive integers. Let m be such an integral ideal of Q, and let m = (m), m ∈ Z, m ≥ 1. We shall g  t p jj , t j ≥ 1 ( p j : distinct write I(m), r(m), etc. for I(m), r(m) etc. Let m = j=1

primes). Then, by the definition, F(m) = {a | a ∈ Q, a > 0, ν P j (a − 1) ≥ t j , 1 ≤ j ≤ g}, and r(m) ↔ F(m) under I ↔ Q+ . Let G(m) = (Z/mZ)∗ ; G(m) is the multiplicative group of residue classes u¯ (= u mod m) such that (u, m) = 1. It is an abelian group of order ϕ(m) (ϕ = Euler’s function). Let u ∈ Z, u ≥ 1, (u, m) = 1. Then the principal ideal (u) is in I(m), and the coset of (u) mod r(m) will be denoted by c(u). Let v ∈ Z, v ≥ 1, (v, m) = 1. Then c(uv) = c(u)c(v). Also, c(u) = c(v) ⇔ ( uv ) ∈ r(m) ⇔ uv ∈ F(m) ⇔ ν p j ( uv − 1) ≥ t j , 1 ≤ j ≤ g ⇔ t u ≡ v mod p jj , 1 ≤ j ≤ g ⇔ u ≡ v mod m. Namely,

78

4 Some Applications

c(u) = c(v) ⇐⇒ u ≡ v mod m. It follows that u(= ¯ u mod m) → c(u) defines an injective homomorphism of G(m) into I(m)/r(m). Let c be any element of I(m)/r(m), and a = (a), a > 0, an ideal in the coset c. Then a = uv , u, v ∈ Z, u, v ≥ 1, (u, m) = (v, m) = 1. One sees immediately that c = c(u)c(v)−1 . Hence the above homomorphism is also surjective, and we obtain an isomorphism G(m) −→ I(m)/r(m),

(4.4)

which maps u¯ to c(u) (u ∈ Z, u ≥ 1, (u, m) = 1). Let w ∈ Z, w ≥ 1, (m, w) = 1, and let c = c(w) ∈ I(m)/r(m). Clearly prime ideals correspond to prime numbers under I → Q+ . Let p = ( p), p = prime number. Then p ∈ c = c(w) ⇔ c( p) = c(w) ⇔ p = w mod m. Hence Theorem 4.1 implies that there exist infinitely many prime numbers p such that p ≡ w mod m. This is the classical theorem of Dirichlet. Let Λ(m) denote the group of all characters of G(m); Λ(m) is also an abelian group of order ϕ(m). For λ ∈ Λ(m), let f λ : Z → C be defined by  λ(u), ¯ for (u, m) = 1, f λ (u) = 0, otherwise. For simplicity, such a map f λ will be also denoted by λ. Let (m 1 ) ⊂ (m 2 ), i.e. m 2 | m 1 (m 1 , m 2 ∈ Z, m 1 , m 2 ≥ 1). Then the natural map Z/m 1 Z → Z/m 2 Z implies a surjective homomorphism G(m 1 ) → G(m 2 ), and hence also an injective homomorphism Λ(m 2 ) −→ Λ(m 1 ). A character λ in Λ(m) is called primitive if λ is not in the image of Λ(m  ) → Λ(m) for any m  ≥ 1, m  ∈ Z, m  | m. The isomorphism (4.4) defines a natural isomorphism Ch(I(m)/r(m)) → Λ(m) so that we have X (m) = X  (m) −→ Ch(I(m)/r(m)) −→ Λ(m), where the first isomorphism is defined in Sect. 4.1. Let χ → ψχm → λ under the above maps. Then one sees easily that χ has the conductor (m) if and only if λ is primitive in Λ(m). Suppose that this is so. Let u ∈ Z, u ≥ 1. If (u, m) = 1, then (u) ¯ = λ(u) because u¯ ↔ c(u) = (u) mod is in I(m), and ψχ ((u)) = ψχm ((u)) = λ(u) r(m) under G(m) → I(m)/r(m). On the other hand, if (u, m) = 1, then ψχ ((u)) = 0, λ(u) = 0, both from the definition of ψχ ((u)) and λ(u). Hence ψχ ((u)) = λ(u)

4.3 Dirichlet’s L-functions

79

for any u ∈ Z, u ≥ 1. It follows that L(s; χ ) =

 −1 1 − ψχ (( p))N (( p))−s ( p)

 −1 1 − λ( p) p −s = p

=

∞ 

λ(u)u −s , Re (s) > 1.

u=1

Therefore L(s; χ ) is also denoted by L(s; λ). We see that when λ ranges over all primitive characters in Λ(m), L(s; λ) gives us all L-functions L(s; χ ) of Q for the characters χ in X (m) with conductor (m). The L-functions of Q were first introduced by Dirichlet in this manner by means of the primitive characters of G(m). Hence they are called Dirichlet’s L-functions. Let χ → λ as above, with λ primitive in Λ(m). Let {m 1 ; s1 } be the signature of χ . Since χ ∈ X  = X , we have s1 = 0 (cf. Sect. 4.1). We shall next show how to determine e = m 1 by means of λ. Let α ∈ J∞ , α = (1, . . . , 1, x), x ∈ R∗ . Then  χ (α) =

x |x|

m 1

 |x|is1 =

x |x|

e .

Put x = −1 so that α = (−1)∞ . Then χ ((−1)∞ ) = (−1)e . Now, χ ((−1)0 · (−1)∞ ) = χ (−1) = 1, χ ((−1)0 · (m − 1)0 ) = χ ((1 − m)0 ) = 1 because (1 − m)0 ∈ U0 (m). Hence χ ((−1)∞ ) = χ ((m − 1)0 ) = ψχ ((m − 1)) = λ(m − 1) = λ(−1). It follows that the signature of χ is {e; 0} where  e=

0, if λ(−1) = 1, 1, if λ(−1) = −1.

In the above, we have implicitly assumed that m > 1. However, if m = 1, then χ ≡ 1, λ ≡ 1 so that e = 0 and λ(−1) = 1. Hence the above formula holds also in this case. With the above e, we then have    m  s+e s+e 2 , A(s; χ ) = , γ (s; χ ) = Γ π 2    m  s+e s+e 2 ξ(s; χ ) = L(s; χ ). Γ π 2 We can also prove that m−1 (−i)e  2πiu W (χ ) = √ λ(u)e m . m u=0

80

4 Some Applications

Problem 4.2 Prove the above formula. Let m ∈ Z, m ≥ 1. Consider   −1 1 − ψχ (( p)) p −s L(s; χ ) = χ

χ∈X (m)

=

p

   p

1 − ψχ (( p)) p

−s

−1 

, Re (s) > 1.

χ

¯ if Suppose that ( p, m) = 1. Then ( p) is in I(m), and ψχ (( p)) = ψχm (( p)) = λ( p) χ → λ under X (m) → Λ(m). Hence       1 − ψχ (( p)) p −s = 1 − λ( p) ¯ p −s . χ∈Λ(m)

λ∈Λ(m)

Let f be the order of p¯ in G(m). Then f is the least positive integer such that p f ≡ 1 2πi mod m, and is a factor of ϕ(m) = #G(m). We put ϕ(m) = f g. Let η = e f . Then it is well-known that when λ ranges over Λ(m), λ( p) ¯ ranges over {ηk | 0 ≤ k < f }, g-times for each. Therefore the above product over Λ(m) equals g  f −1   = (1 − p − f s )g . 1 − ηk p −s k=0 2πi

Let K be the cyclotomic field of mth roots of unity, i.e., K = Q(e m ). Then the ideal ( p) is decomposed in K into the product of prime ideals as follows: ( p) = p1 · · · pg ,

N (pk ) = p f , k = 1, . . . , g;

here f and g are the integers defined in the above. It follows that  p| p

1 − N (p)

−s



g    = 1 − N (pk )−s k=1

= (1 − p − f s )g so that

     1 − ψχ (( p)) p −s = 1 − N (p)−s . χ∈X (m)

p| p

This is proved for any prime number p which is prime to m. However, we can obtain the same formula also for p which divides m. (The proof is similar, but is slightly more complicated). Therefore

4.3 Dirichlet’s L-functions

81



L(s; χ ) =

χ∈X (m)

 −1 1 − N (p)−s p p| p

 −1 1 − N (p)−s = p

= ζ K (s), Re (s) > 1, and we have the following Theorem 4.2 Let K be the cyclotomic field of m-th roots of unity, m ≥ 1. Then

ζ K (s) =



L(s; χ ), s ∈ C.

χ∈X (m)

The product is taken our all Dirichlet’s L-functions with characters χ in X (m). The above theorem has many interesting consequences. We know that both ζ K (s) and L(s; 1) = ζQ (s) have a simple pole at s = 1 and that L(s; χ ), χ ≡ 1, is holomorphic at s = 1. Hence it follows from the above that L(1; χ ) = 0 for χ ≡ 1 (cf. Sect. 4.2); for, otherwise, the zero of L(s; χ ) at s = 1 would cancel the pole of L(s; 1) at s = 1, and ζ K (s) would be holomorphic at s = 1. Furthermore, since the residue of ζQ (s) at s = 1 is 1, it also follows from the above that the residue of ζ K (s) at s = 1 is equal to  L(1; χ ). χ∈X (m) χ≡1

Hence (cf. Sect. 3.6) we have √  w d h = r +r r L(1; χ ) 1 2 2 2 π R χ∈X (m) χ≡1

for the class number h of K . In the next section, we shall compute the value of L(1; χ ) (χ ≡ 1), and obtain the classical class number formula for the cyclotomic field K (with m a prime number). Actually, the equality in Theorem 4.2 is a special case of the following important general result in class field theory. Let F be any number field and let Y be any finite subgroup of the group X of Hecke characters of F. Since X  is the torsion subgroup of X , Y is contained in X  . As stated in Sect. 4.1, X  may be identified with the character group of G, the Galois group of the maximal abelian extension A over F. Let H be the closed subgroup of all σ in G such that χ (σ ) = 1 for χ in Y , and let K be the field, F ⊂ K ⊂ A, corresponding to the subgroup H of G. Then K /F is a finite abelian extension with Galois group G/H . For such an extension K of F, we then have  ζ K (s) = L(s; χ ), s ∈ C. χ∈Y

82

4 Some Applications

∼ G/H (not Note that Y may be considered as the character group of G/H so that Y = canonically) and [K : F] = [G : H ] = order of Y . In Theorem 4.2, we have F = Q and Y = X (m).

4.4 The Class Number Formula for Cyclotomic Fields We first make a remark on the regulator of a number field F. Let ε1 , . . . , εr −1 (r = r1 + r2 ) be any r − 1 units of F. Define

⎛

e1 log |ε1(1)

⎜ ..

⎜ . R(ε1 , . . . , εr −1 ) = det ⎜

⎝e log |ε(1) | 1

r −1 e1

n

⎞

· · · er log |ε1(r ) |

⎟

.. .. ⎟

. . ⎟ , (r ) ⎠

· · · er log |εr −1 |

er

··· n

where ek = 1 or 2 accordding as 1 ≤ k ≤ r1 or r1 + 1 ≤ k ≤ r . If ε1 , …, εr −1 form a basis of E/W , then R = R(ε1 , . . . , εr −1 ) is the regulator of F. Therefore, in such a case, R(ε1 , . . . , εr −1 ) is independent of ε1 , …, εr −1 . In general, for any units ε1 , . . . , εm , m ≥ 1, let E(ε1 , . . . , εm ) denote the subgroup of E generated by ε1 , …, εm . Then R(ε1 , . . . , εr −1 ) = 0 if and only if E(ε1 , . . . , εr −1 ) (or E(ε1 , . . . , εr −1 )W ) has a finite index in E, and in that case, we have   R(ε1 , . . . , εr −1 )/R = E : E(ε1 , . . . , εr −1 )W . Now, let l be a fixed prime number, l > 2, and let K denote the cyclotomic field of l-th roots of unity; 2πi K = Q(ζ ), ζ = e l . Let K 0 = Q(ζ + ζ ); K 0 is the maximal real subfield of K contained in K . We put n = [K : Q] = ϕ(l) = l − 1, l −1 , n 0 = [K 0 : Q] = 2 h = class number of K , h 0 = class number of K 0 , R = regulator of K , R0 = regulator of K 0 , d = |discriminant of K | = l n−1 ,

4.4 The Class Number Formula for Cyclotomic Fields

83

d0 = |discriminant of K 0 | = l n 0 −1 , E = group of units in K , E 0 = group of units in K 0 , W = group of roots of unity in K , W0 = group of roots of unity in K 0 . We can prove that W = {±ζ a }, 0 ≤ a < l, so that w = order of W = 2l. It is obvious that W0 = {±1} and w0 = order of W0 = 2. Let  √ n−1  √ g = (2π )n 0 R/2l l = 2r1 +r2 π r2 R/w d with r1 = 0, r2 = n 0   √ n 0 −1  g0 = 2n 0 R0 /2 l = 2r1 +r2 π r2 R0 /w0 d0 with r1 = n 0 , r2 = 0 so that gh (resp. g0 h 0 ) is the residue of ζ K (s) (resp. ζ K 0 (s)) at s = 1. It is known that h 0 is a factor of h: h = h 1 h 0 , h 1 ∈ Z. Sometimes h 1 is called the first factor of the class number h, and h 0 the second factor of h. It can also be proved that E = {ζ a } × E 0 . It follows in particular that if ε1 , …, εn 0 −1 (r = r1 + r2 = n 0 for both K and K 0 ) form a basis of E 0 /W0 , then they also form a basis of E/W . We then see immediately from the definition of regulators that R = 2n 0 −1 R0 so that √ n0 g/g0 = π n 0 2n 0 −1 /l l . For each integer a, prime to l, there exists a unique Galois automorphism σa of K /Q such that σa (ζ ) = ζ a . Clearly σa depends only upon the residue class a of a mod l. The map a → σa then defines an isomorphism G(l) ∼ = G = Galois group of K /Q. The Galois group of K /K 0 corresponds in this isomorphism to the subgroup {±1}. Hence

84

4 Some Applications

G(l)/{±1} ∼ = G 0 = Galois group of K 0 /Q. We fix an integer t (a primitive root mod l) such that t is a generator of G(l), and we put σ = σt so that G = {σ k }, 0 ≤ k < n. For any α in K , we then denote the conjugates of α by α (k) = σ k (α), 0 ≤ k < n = l − 1. Let l be the principal ideal of K generated by 1 − ζ : l = (1 − ζ ). Then l is a prime ideal of K , and (l) = ln . Hence σ (l) = l, and σ (1 − ζ )/(1 − ζ ) = (1 − ζ t )/(1 − ζ ) is a unit of K . Let η=ζ

1−t 2

(1 − ζ t )/(1 − ζ ) = (ζ 2 − ζ − 2 )/(ζ 2 − ζ − 2 ). t

t

1

1

1

(Note that ζ 2 is in W because ζ has order l, (l, 2) = 1). It follows from the above that η is a unit, in fact, a real unit: η ∈ E 0 . It is called the circular unit of K (for fixed t). Since G(l) = (Z/lZ)∗ is a cyclic group of order n = l − 1, so is the character group Λ(l) (and also X  (l) = X (l)).  Hence Λ(l) has a unique character ρ of order 2: ρ 2 ≡ 1, ρ ≡ 1. Since a → al , a ∈ Z, is such a character, we have ρ(a) =

a  l

, a ∈ Z.

Clearly ρ takes only real values (namely, 0, ±1). However, if conversely λ is a character in Λ(l) taking only real values, then λ(a) = ±1 for (a, l) = 1. Hence λ2 ≡ 1, and we see that either λ ≡ 1 or λ = ρ in the above. By this reason, we call ρ the real character in Λ(l). Let Λ0 (l) denote the set of all λ in Λ(l) such that λ(−1) = 1. Then Λ0 (l) is a in Λ(l), and it may be considered as the group of all subgroup of order n 0 = l−1 2   ¯ Since ρ(−1) = −1 = (−1) l−1 2 , we see that ρ ∈ Λ (l) or characters of G(l)/{±1}. 0 l ρ∈ / Λ0 (l) according as l ≡ 1 mod 4 or l ≡ 3 mod 4. Let 2πia ω(a) = ζ a = e l , a ∈ Z. Then ω(a) depends only upon the coset a¯ of a mod l, and it defines a character of the additive group of the finite ring (field) R = Z/lZ. For any λ in Λ(l), we put v(λ) =

 a

ω(a)λ(a) =

l−1  a=0

λ(a)ζ a .

4.4 The Class Number Formula for Cyclotomic Fields

85

v(λ) is nothing but the Gaussian sum G(ω, λ) on R defined in Sect. 2.3, and we see immediately from the lemmas proved there that 

λ(a)ζ ua = λ(u)v(λ),

(u, l) = 1 or λ ≡ 1,

(4.5)

a

v(λ) = 0, √ |v(λ)| = l, It follows that v(λ) =



λ = 1, λ ≡ 1.

λ(a)ζ −a = λ(−1)v(λ).

(4.6)

a

In particular, v(ρ) Hence v(ρ)2 = ρ(−1)|v(ρ)|2 = ρ(−1) · l = ±l, √ √ = ρ(−1)v(ρ). and v(ρ) = ± l or ±i l according√as l ≡ 1√mod 4 or l ≡ 3 mod 4. It is not quite easy to determine the sign ± before l and i l. One can show however that √ l, l ≡ 1 mod 4, v(ρ) = √ i l, l ≡ 3 mod 4,

√ ( l > 0).

We know from Sect. 4.4 that   L(1; χ ) = L(1; λ). gh = χ∈X (l) χ≡1

λ∈Λ(l) λ≡1

Applying a similar argument to K 0 (or, from the general theorem stated at the end of Sect. 4.4), we also see that 

g0 h 0 =

L(1; λ).

λ∈Λ0 (l) λ≡1

To compute the value of L(1; λ) for λ ≡ 1, we first prove the following, Lemma 4.4 Let ζ be any root of unity, ζ = 1 (not necessarily ζ = e f (s) =

∞ 

2πi l

). Let

ζ m m −s , s > 0.

m=1

For any ε > 0, the above series converges uniformly for s ≥ ε so that f (s) defines a continuous function of s for s > 0. In particular lim f (s) = f (1) =

s→1

∞  m=1

ζ m m −1 = − log(1 − ζ ),

86

4 Some Applications

where log(1 − ζ ) = log |1 − ζ | + i arg(1 − ζ ), −

π π < arg(1 − ζ ) < . 2 2

 Proof Let S(a) = am=1 ζ m for a ≥ 1, and S(0) = 0. If ζ is an N -th root of unity, a+N  ζ = 1, then ζ m = 0. Hence S(a) = S(b) for a ≡ b mod N , and one obtains m=a

immediately that |S(a)| ≤ N , a ∈ Z. Let s ≥ ε, b ≥ a. Then b 

ζ m m −s =

m=a

b 

(S(m) − S(m − 1)) m −s

m=a

= −S(a − 1)a −s + S(b)b−s +

b−1 

  S(m) m −s − (m + 1)−s .

m=a

Hence it follows from the above that

b

b−1



  

m −s

−s −s ζ m ≤ Na + Nb + N S(m) m −s − (m + 1)−s

m=a

m=a = 2N a −s ≤ 2N a −s −→ 0,

for a −→ +∞

This proves the uniform convergence. The fact that

∞ 

ζ m m −1 = − log(1 − ζ ) is

m=1

well-known in the function theory. Now, let λ ≡ 1, s > 1, and ζ = e

2πi l

. Using

 l−1 1  (a−m)u 1, m ≡ a mod l, ζ = l u=0 0, m ≡ a mod l, one obtains that L(s; λ) =

∞ 

λ(m)m −s =

m=1

=

l−1  a=0

l−1  a=0

λ(a)

∞ 

1 l m=1

l−1  u=0

λ(a)

 m≡a mod p

ζ (a−m)u m −s

m −s

4.4 The Class Number Formula for Cyclotomic Fields

87

=

l−1 l−1 ∞  1  λ(a)ζ au ζ −mu m −s l u=0 a=0 m=1

=

l−1 ∞  v(λ)  λ(u) ζ −mu m −s l u=0 m=1

(by (4.5))

=

l−1 ∞  v(λ)  λ(u) ζ −mu m −s l u=1 m=1

(∵ λ(0) = 0).

Since ζ −u = 1 for 1 ≤ u ≤ p − 1, we obtain from the lemma that v(λ)  λ(u) log(1 − ζ −u ) l u=1 l−1

L(1; λ) = lim L(s; λ) = − s→1

  v(λ)  λ(u) log(1 − ζ −u ) + λ(−1) log(1 − ζ u ) . 2l u=1 l−1

=−

Let λ ∈ Λ0 (l) so that λ(−1) = 1. Let α = (1 − ζ )(1 − ζ −1 ). Then   v(λ)  λ(u) log(1 − ζ −u ) + log(1 − ζ u ) 2l u=1 l−1

L(1; λ) = −

v(λ)  λ(u) log σu (α) =− 2l u=1 l−1

v(λ)  λ(t)k log α (k) , 2l k=1 l−1

=−

where α (k) = σ k (α) = σt k (α). Since α is contained in K 0 , we have σ n 0 (α) = α, α (n 0 +k) = α (k) . Also, λ(t)n 0 = λ(t n 0 ) = λ(−1) = 1. Hence L(1; λ) = −

n 0 −1 v(λ)  λ(t)k log α (k) . l k=0

It follows that h 0 = g0−1

 λ∈Λ0 (l) λ≡1



 |L(1; λ)| L(1; λ) = g0−1

λ

88

4 Some Applications

n −1

 0

1 n 0 −1  



= λ(t)k log α (k)

√



l k=0

n −1

0

 −1  



= 2n 0 −1 R0 λ(t)k log α (k) .





2n 0 −1 R0 √ n 0 −1 l

−1 

k=0

2πi

m

When λ ranges over the character group Λ0 (l), λ(t) takes the values e n0 , m = 0, 1, . . . , n 0 − 1. Hence by a well-known theorem on determinants, we have

⎛ log α (0) log α (1)







−1 n 0 ⎜

⎜ log α (1) log α (2)

  k (k)

λ(t) log α .. ..

=

det ⎜

. .

⎝

λ∈Λ0 (l) k=0

log α (n 0 −1) log α (0) Since η = ζ

1−t 2

1−ζ t 1−ζ

, η(k) = ζ

(1−t)t k 2

⎞

· · · log α (n 0 −1)



· · · log α (0) ⎟ ⎟

. ⎟ . .. .. ⎠

.

· · · log α (n 0 −2)

k+1

1−ζ t 1−ζ t k

,

t k+1 −t k+1

(k) 2

η = (1 − ζ k )(1 − ζ k ) (1 − ζ t )(1 − ζ −t ) (k+1) α = (k) α

Therefore the above determinant equals ⎛

2 log |η(1) | · · · ⎜ 2 log |η(2) | · · · ⎜ ⎜ .. .. det ⎜ . . ⎜ ⎝2 log |η(n 0 −2) | 2 log |η(n 0 −1) | · · · log α (0) ··· log α (n 0 −1) Using

n 0 −1

2 log |η(0) | 2 log |η(1) | .. .

⎞ 2 log |η(n 0 −1) | 2 log |η(0) | ⎟ ⎟ ⎟ .. ⎟. . ⎟ (n 0 −3) ⎠ 2 log |η | log α (n 0 −2)

log |η(k) | = log N K 0 /Q (η) = log 1 = 0 (because η is a unit), we then see

k=0

that the determinant is again equal to the following: ⎛ 0 log |η(1) | · · · ⎜0 log |η(2) | · · · n −1 0 ⎜  ⎜ .. .. log α (k) det ⎜ ... ±2n 0 −1 . . ⎜ k=0 ⎝0 log |η(n 0 −1) | · · · 1 log α (0) · · ·

⎞ log |η(n 0 −1) | log |η(0) | ⎟ ⎟ ⎟ .. ⎟ . ⎟ log |η(n 0 −3) |⎠ log α (n 0 −2)

4.4 The Class Number Formula for Cyclotomic Fields

89

⎞ ⎛ 0 log |η(1) | · · · log |η(n 0 −1) | ⎜0 log |η(2) | · · · log |η(0) | ⎟ n −1 0 ⎟ ⎜  ⎟ ⎜ .. .. .. = ±2n 0 −1 log α (k) det ⎜ ... ⎟ . . . ⎟ ⎜ k=0 ⎝0 log |η(n 0 −1) | · · · log |η(n 0 −3) |⎠ 1 0 ··· 0 ⎞ ⎛ log |η(0) | log |η(1) | · · · log |η(n 0 −1) | ⎜ log |η(1) | n −1 log |η(2) | · · · log |η(0) | ⎟ 0 ⎟ ⎜  ⎟ ⎜ .. .. .. .. n 0 −1 (k) = ±2 log α det ⎜ ⎟ . . . . ⎟ ⎜ k=0 ⎝log |η(n 0 −2) | log |η(n 0 −1) | · · · log |η(n 0 −3) |⎠ 1 1 1 ··· n0 n0 n0 n −1 0  n 0 −1 (k) = ±2 log α R(η(0) , . . . , η(n 0 −2) ). k=0

If λ ≡ 1, then n 0 −1

λ(t)k log α (k) =

k=0

n 0 −1

log α (k) = log N K 0 /Q (α)

k=0

= log N K /Q (1 − ζ ) = log p = 0. Hence we obtain from the above that

−1

0



n k (k)

λ(t) log α = 2n 0 −1 R(η(0) , . . . , η(n 0 −2) ),



λ∈Λ0 (l) k=0 λ≡1

and consequently that h0 =

R(η(0) , . . . , η(n 0 −2) ) . R

Since η(0) · · · η(n 0 −1) = N K 0 /Q (η) = 1, E(η(0) , . . . , η(n 0 −2) ) is the subgroup of E 0 generated by all conjugates η(k) of η, and E(η(0) , . . . , η(n 0 −2) )W0 is the subgroup of E 0 generated by the conjugates ±η(k) of ±η. By the remark stated at the beginning of this section, we have the following formula for the class number h 0 of K 0 : h 0 = [E 0 : H ]; here E 0 is the group of all units in K 0 , and H is the subgroup of E 0 generated by the conjugates ±η(k) of ±η. We shall next compute h 1 = hh0 from

90

4 Some Applications

h1 =

h g0  = L(1; λ) h0 g λ∈Λ(l) λ∈Λ / 0 (l)

√ n0  l l L(1; λ). = n n −1 π 02 0 λ Let λ ∈ Λ(l), λ ∈ / Λ0 (l) so that λ(−1) = −1. Then   v(λ)  λ(u) log(1 − ζ −u ) − log(1 − ζ u ) 2l u=1 l−1

L(1; λ) = −

v(λ)  λ(u) · 2i arg(1 − ζ −u ). 2l u=1 l−1

=− However,

u 1 2π − 2π 2 l u = π −π . l

arg(1 − ζ −u ) =

u

O•

1 −u

Since

l−1 

λ(u) = 0 for λ = 1 (or λ = 1), we obtain

u=1

πiv(λ)  uλ(u). l 2 u=1 l−1

L(1; λ) = Therefore

√ n0 l−1  l l π n0 i n0  h 1 = n n −1 · 2n · v(λ) · uλ(u). π 02 0 l 0 λ λ u=1

Let l ≡ 1 mod 4, n 0 = l−1 ≡ 0 mod 2. Then ρ ∈ Λ0 (l), and λ = λ for λ ∈ / Λ0 (l). 2 Hence the characters in the above product consist of n20 pairs {λ, λ} with λ ∈ / Λ0 (l). Since v(λ) = −v(λ) by (4.6), we have v(λ)v(λ) = −|v(λ)|2 = −l. Hence

4.4 The Class Number Formula for Cyclotomic Fields



91

√ n0 n0 v(λ) = (−l) 2 = i n 0 l .

λ

≡ 1 mod 2. Then ρ ∈ / Λ0 (l), and the characters in the Let l ≡ 3 mod 4, n 0 = l−1 2 √ above product consist of the real character ρ and n 02−1 pairs {λ, λ}. Since v(ρ) = i l, we have  √ √ n0 n 0 −1 v(λ) = i l(−l) 2 = i n 0 l . λ

Therefore, in either case, √ n0 l l π n0 i n0  (−1)n 0 · 2n · v(λ) = , n n −1 π 02 0 l 0 (2l)n 0 −1 λ and we obtain the following formula for h 1 = hh0 (the first factor of the class number h of K ):  l−1  h 1  h1 = = 2l uλ(u) . − h0 2l u=1 λ∈Λ(l) λ(−1)=−1

By the same method, we can obtain similar class number formulas for more general types of cyclotomic√ fields as well as their subfields. For example, let l ≡ 3 mod 4, l > 3. Then F = Q( −l) is a subfield of K in the above. Let h  be the class number of F, and let π g = √ . l Then

namely,

g  h  = L(1; ρ), √ l L(1; ρ). h = π 

However, we have seen in the above that √ l−1 l−1 πiv(ρ)  πi(i l)   u  L(1; ρ) = u. uρ(u) = l 2 u=1 l2 l u=1 Hence

1  u  u. l u=1 l l−1

h = −

92

4 Some Applications

For l = 3, we have g  = 3π√l so that we have to multiply the right-hand side of the above by 3. (We then obtain√ h  = 1). Let l ≡ 1 mod 4. Then Q( l) is contained √ in the above cyclotomic field K , and we can obtain a class number formula for Q( l) similar to the formula h 0 = [E 0 : H ] for K 0 .

Bibliography

1. E. Artin, The theory of algebraic numbers, Notes by Gerhald Würges from lectures held at the Mathematisches Institut, Göttingen, Germany in the Winter-Semester, 1956/57, Translated and distributed by George Sticker. Göttingen 172 pages, 1959. 2. E. Hecke, Vorlesungen über die Theorie der algebraischen Zahlen. Chelsea, 1948.

Reference added by Compiling Editors 3. J. Tate, Fourier analysis in number fields and Hecke’s zeta-functions, in “Algebraic Number Theory” Edited by J. Cassels and A. Frohlich, pp. 305–347, Thompson Washington, D.C., 1967.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2019 K. Iwasawa, Hecke’s L-functions, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-981-13-9495-9

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