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*Table of contents : On Congruence Monodromy .........Page 0Contents......Page 2Preface to Volume 1.......Page 5Preface to Volume 2.......Page 6Part 1. The group $\Gamma$ .........Page 10Part 2 Detailed study .........Page 37Introduction to Part 1 .........Page 71Part 1. The $G_{\mathfrak{p}}$ .........Page 73Part 2. Full $G_{\mathfrak{p}}$ .........Page 91Introduction to Part 3A .........Page 114Part 3A. The canonical .........Page 117Part 3B. Unique existence .........Page 134Part 1. Some properties .........Page 144Part 1. Examples of $\Gamma.$......Page 160Part 1. Elliptic modular .........Page 165Part 2. Non-abelian classfields .........Page 174Supplements.......Page 213Bibliography......Page 221Author's Notes (2008)......Page 223Additional References......Page 235*

On Congruence Monodromy Problems Yasutaka Ihara

Reproduction of the Lecture Notes: Volume 1 (1968), Volume 2(1969) (University of Tokyo) With Author’s Notes (2008)

iii

Contents Preface to Volume 1.

xi

Preface to Volume 2.

xiii

General Introduction.

xv

Notations.

xvii

Chapter 1. Part 1. The group and its -function. \S \S 1– 5. Discrete subgroup . \S \S 6- 8. The function of . \S \S 9-13. Lemmas for the proof of Theorem 1. The proof of Theorem 1 assuming Lemmas 2, 3. \S 14. \S \S 15-19. Proofs ofLemmas 2, 3. Regular cycles on . \S 20. . \S \S 21-23. Estimation of the roots of Concluding remarks on Chapter 1, Part 1. \S 24. Part 2. Detailed study of elements of with parabolic and elliptic real parts; the general fomlula for . \S \S 25-28. Study of elements of with parabolic real parts. \S \S 29-34. Study of elements of with elliptic real parts. \S \S 35-38. The function of in the general case.

28 29 37 53

Chapter2.

63

$\Gamma$

$\zeta$

$\Gamma$

$\zeta$

$\Gamma$

$\Gamma_{R}^{0}\backslash \mathfrak{H}$

$\zeta_{\Gamma}(u)$

1 1 1 5 7 13 15 24 24 27

$\Gamma$

$\zeta_{\Gamma}(u)$

$\Gamma$

$\Gamma$

$\zeta$

$\Gamma$

Introduction to Part 1 and Part 2. Part 1. The -fields over C. \S \S 1– 4. The -fields. \S \S 5-10. Analytic consﬀuction of -fields over C. \S \S 11--17. The full automorphism group of over C. Part 2. Full -subfields over algebraic number fields. Main results. \S 18. \S \S 19-20. Reducing Theorem 5 to Theorem 4. \S \S 21-26. Preliminaries for the proof of Theorem 4. \S \S 27-30. More lemmas. ProofofTheorem4 (Conclusion). \S 31. \S \S 32-34. Variations of Theorems 4, 5. . \S \S 35-36. The fields $h$ and $G_{\mathfrak{p}}$

$G_{\mathfrak{p}}$

$G_{\mathfrak{p}}$

$L$

$G_{p}$

$F=Q((tr\gamma_{R})^{2}|\gamma_{R}\in\Gamma_{R})$

vii

63 65 65 69 77 83 83 83 85 90 96 98 103

viii

Introduction to Part 3A and Part 3B . Part 3A. The canonical -operator and the canonical class of linear diﬀerential equations of second order on algebraic function field of one variable over , and their algebraic characterizations when is ariﬃmetic”. \S \S 37-38. The -operators. \S \S 39-42. The canonical -operator on algebraic fimction field of one variable over , and its algebraic characterization in ample

106

$S$

$C$

$L$

$L$

$S$

109 109

$S$

$C$

(arithmetic)

cases.

110

\S \S 43-44. The canonical class of linear diﬀerential equations of second order on algebraic fimction fields over , and its algebraic characterization in ample(arithmetic) cases. Part 3B. Unique existence of an invariant -operator on “arithmetic” algebraic function fields (including -fields) over any field of characteristic zero. Unique existence of invariant -operator on ample (arithmetic) \S 45. $L/k$ . and Theorem 10. \S \S 46-47. Proofs ofLemma Some corollaries and applications of Theorem 10. \S 48. $C$

120

$S$

$G_{\mathfrak{p}}$

126

$S$

$14_{k}$

126 128 133

Chapter3. Part 1. Some properties of . and its consequences. \S \S 1-7. The vanishing of ; \S \S 8-9. Applications of Theorem 3 the deformation variety. . \S \S 10-14. Study of ;the field

137 137 138 144 146

Chapter4. Part 1. Examples of . \S \S 1-3. Examples of . \S \S 4-5. That $h$ contamins F.

153 153 153 156

Chapter5. Part 1. Elliptic modular functions $mod p$ and $\Gamma=PSL_{2}(Z^{(p)})$ . \S \S 1-5. Elliptic modular functions $mod p$ and $\Gamma=PSL_{2}(Z^{(p)})$ . \S \S 6-7. Deuring’sresults. ProofofTheorems 1, 1’. \S 8. \S \S 9-10. Acorollary and an announcement ofgeneralizations. Part 2. Non-abelian classfields attached to subgroups of $\Gamma=PSL_{2}(Z^{(p)})$ with fimite indices. \S \S 11-14. Preliminaries on elliptic curves; results of Igusa and Shimura. . \S \S 15-16. The group and the extension \S \S 17-21. Afundamental theorem. \S \S 22-25. Decomposition of ordinary prime divisors of in . \S \S 26-27. Decomposition of supersingular prime divisors of in .

159 159 159 162 164 166

$\Gamma$

$H^{1}(\Gamma_{R},\rho_{n})$

$F=Q((ff\gamma_{R})^{2}|\gamma_{R}\in\Gamma_{R})$

$V_{0}$

$\Gamma$

$\Gamma$

$\Gamma^{*}$

$\mathfrak{R}/\overline{k}$

$\overline{k}$

$\mathfrak{R}$

$\overline{k}$

$\mathfrak{R}$

168 170 175 179 186

190

ix

CHAPTER . CONTENTS $0$

Decomposition ofthe infinite prime divisor of in . \S 28. \S \S 29-30. Reformulation in terms of non-abelian classfields; Main Theorems ( -1)\sim ( -3), and Conjecture . \S \S 31-33. Supplements to Main Theorems and to Conjecture . $\overline{k}$

$\mathfrak{R}$

$\Gamma^{*}$

$\Gamma^{*}$

$\Gamma^{*}$

$\Gamma^{*}$

193 195 198

Supplements.

207

Bibliography.

215

Author’s Notes (2008).

217

Additional References.

229

Preface to Volume 1. This is the first volume of a projected series ofmonographs to appear under the unified title On Congmence Monodromy Problems.” The main content ofthis volume (and a part of the next one) is based on my lectures given at Princeton University in the spring teml of 1967, and at the University of Tokyo in the fall and the winter term, 1967-68. A brief summary of the main content is given without proofs in [18], which will appear in the Joumal of Mathematical Society ofJapan 20-1, 1968. My aim in this series is to solve the problems presented in the title, which are described briefly in the introduction and in more detail in [18]. To this end, several diﬀerent approaches are required, and they are introduced separately in diﬀerent chapters. Therefore, in this volume where each approach is still in its beginning stage, each chapter is almost independent of another. On the other hand, each chapter will be continued on to the successive volumes, so that we may present any further development of the approach introduced in that chapter. Brief explanations on the relations between our problems and results are given in [18]. Here, I wish to express my sincere thanks to the professors of the above mentioned universities for giving me the occasions to lecture on this nascent subject, and to the professors and colleagues of the universities who attended my lectures with patience and gave me several useful remarks. I would also like to acknowledge some valuable remarks given by Professors R. P. Langlands and H. Shimizu (some ofwhich corrected my errors), and encouragement and stimulation provided by Professors Y Kawada and A. Weil. Now, my deepest acknowledgment is to Professor G. Shimura for his interest in this subject, constant encouragement and most valuable discussions. I regret that in this volume I could not make clear the significance ofhis results [32] on Shimura-curves 2 and their relations to our problems. Actually, there are almost no relations at this stage of our problems and results; however, at the following stages of Chapter 4, his results [32] will play an essential role. At any rate, I have received much encouragement and stimulation (and entertained some filrther conjectures) from his results. Finally, I am sincerely thankful to Miss C. Sagawa in the University of Tokyo for typing this lengthy monograph. Y IHARA Tokyo, Japan, Febmary 1968, l This

is mainly based on the former. However, Chap. 2 \S \S 35, 36; Chap. 3 \S 10-\S 14; and Chap. 4 \S \S 4, 5 are newly added, and Chap. 2 was moreover considerably revised. In Tokyo, Chap. 1 and Chap. 5 (including the forthcoming Part 2) were given with more expositions. 2 I.e., algebraic curves umiformized by automorphic functions attached to quatemion algebras over totally real algebraic number fields, whose beautiful arithmetic properties have been discovered by Shimura (in [32]); before him, nothing was known about them except in the elliptic modular case. xi

Preface to Volume 2. This volume consists of continuations of the following three chapters: Chapter 1 (Part 2), Chapter 2 (Parts 3A, 3B)3 Chapter 5 (Part 2)4 The main content of each chapter is summarized at the beginning of each chapter (in (of Part 1) to this volume). In Chapter 1 (Part 2), we generalize the calculation of the cases where the quotient may not be compact and may not be torsion-free, after some detailed studies of elements of with parabolic and elliptic real parts. In Chapter 2 (Parts 3A, 3B), we consider certain diﬀerential equation whose solutions define the given unifonnization of algebraic function field, and show that such an equation can be uniquely characterized algebraically and can be “carried over” to algebraic function fields with arbitrary constant fields of characteristic zero, provided the given algebraic function field is“arithmetic” (or ”ample”; see Chapter 2). In Chapter 5 (Part 2), we give a detailed formulation and proofofthe formerly announced results on non-abelian classfields” over ( a variable over ), attached to the group $\Gamma=PSL_{2}(Z^{(p)})$ . (To each normal subgroup of with finite index, a finite Galois extension $K’$ of $K$ called the -classfield” is defined, and ). Here, the author wishes to express his sincere gratitude, in addition to those persons mentioned in the preface to Volume 1, to Professor J. P. Serre for valuable and stimulating discussions and remarks given during his stay in Tokyo in October 1968. Y. IHARA Tokyo, May 1969, $\zeta_{\Gamma}(u)$

$\Gamma$

$ G/\Gamma$

$\Gamma$

$K=F_{p^{2}}(\overline{j)}$

$\overline{j:}$

$F_{p^{2}}$

$\Gamma’$

$\Gamma’$

$\Gamma$

$\cdots$

3

Talked in a seminar at the University of Tokyo in December 1968, and lectured in a short course at Osaka University in May, 1969. 4 Lectured at the University of Tokyo in Spring Term, 1968 (except \S \S 29\sim 33, which were added in making this note). xiii

General Introduction. Here, only a general introduction is given. The content of each chapter is summarized in the introduction to each chapter.

THE PROBLEMS.5 Let $G=PSL_{2}(R)\times PSL_{2}(k_{\mathfrak{p}})$ , where $R$ and are the real number field and a -adic number field with respectively, and $PSL_{2}=SL_{2}\prime\pm 1$ . Let be a torsion-free discrete subgroup of $G$ with compact quotient, having a dense image of projection in each component of $G$ . Our subject is such a discrete subgroup . This study was motivated by the following series of conjectures which were suggested by our previous work [15]. Since our group is essentially non-abelian (see Chapter 3, Theorem 2 in \S 6), the readers will see that, by our conjectures, would describe a”nonabelian class field theory” over an algebraic function field of one variable with finite constant field We would like to call the problems of determining the validity of these conjectures, the congruence monodromy problems. $k_{\mathfrak{p}}$

$N\mathfrak{p}=q$

$\mathfrak{p}$

$\Gamma$

$\Gamma$

$\Gamma$

$\Gamma$

$F_{q^{2}}.$

CONJECTURES. With each , we can associate an algebraic function field $K$ of one consisting variable with finite constant field and with genus $g\geq 2$ , and a finite set of $(q-1)(g-1)$ prime divisors of $K$ of degree one over satisfying the following properties. Here the elements of are called the exceptional prime divisors, while all $K$ other prime divisors of are called the ordinary prime divisors. $\Gamma$

$\mathfrak{S}(K7$

$F_{q^{2}}$

$F_{q^{2}}$

$\mathfrak{S}(K)$

CONJECTURE 1. The ordinary prime divisors $P$ of $K$ are in one-to-one correspondence with the pairs of mutually inverse primitive elliptic conjugacy classes of (See Chapter 1, \S \S 1-12 for the definitions). $\Gamma$

$\{\gamma_{P}\}_{\Gamma}^{\pm}$

CONJECTURE 2. The finite unramified extensions $K$ over $K$ , in which all $(q-1)(g-1)$ exceptional prime divisors of $K$ are decomposed completely, are in one-to-one correspondence with the subgroups of with finite indices. Moreover, this one-to-one correspondence satisfies the Galois theory. $\Gamma’$

$\Gamma$

CONJECTURE 3. The law of decomposition of ordinary prime divisors of $K$ in $K$ is described by the corresponding and . Namely, decompose the -conjugacy class into a disjoint union of -conjugacy classes: $P$

$\Gamma’$

$\Gamma$

$\{\gamma_{P}\}_{\Gamma}^{\pm}$

$\Gamma’$

$\{\gamma_{P}\}_{\Gamma}$

$\{\gamma_{P}\}_{\Gamma}=\{\gamma_{P,1}\}_{\Gamma’}\cup\cdots\cup\{\gamma_{P,t}\}_{\Gamma’},$

and for each , let we have $i$

is contained in . Then be the smallest positive integer such that , and our conjecture asserts that the decomposition of $P$ in $K$ is $\Gamma’$

$f_{i}$

$\sqrt{Pt},i$

$\sum_{i=1}^{t}f=(\Gamma:\Gamma’)$

$5_{Here}$

,

we shall reproduce a palt ofthe introduction ofmy paper [18]. As for the details, cf. xv

[18] \S 3.

xvi

of type $P=P_{1}P_{2}\cdots P_{t},$

where $P_{i}(1\leq i\leq t)$ are prime divisors of $K$ with relative degrees $f_{i}(1\leq i\leq t)$ respectively. As mentioned in the preface, we shall attack the problems from several diﬀerent approaches. The results obtained are still far from the solution of the problems, but are very encouraging. For the summary of the contents, the readers are asked to refer to the introduction given at the beginning of each chapter. There is some interdependence between the five chapters, but they are not serious, and references are explicitly indicated. Therefore, each chapter can be read independently without the knowledge of the others. The only exception is Chapter 1, \S \S 1 3, which is presupposed throughout the volume. $\sim$

Notations. (i)

: the field of rational numbers; : its algebraic closure. $R$ : the field of real numbers; : the field of complex numbers; Re,Im: the real and the imaginary parts. $\overline{Q}$

$Q$

$C$

(ii)

: the ring of rational integers. , where is a prime number.

$Z$

$Z^{(p)}=\bigcup_{n=0}^{\infty}\frac{1}{p^{l}}Z$

(iii)

: a -adic number field; : the ring of integers of ; ; : the maximal ideal of : the multiplicative group : the normalized additive valuation of : the -adic number field; : the ring of integers $ofQ_{p}.$

$k_{\mathfrak{p}}$

$O_{\mathfrak{p}}$

$\mathfrak{p}$

$\mathfrak{p}$

$k_{\mathfrak{p}}$

$O_{\mathfrak{p}}-\mathfrak{p}$

$O_{\mathfrak{p}};\mathcal{U}_{\mathfrak{p}}$

$q=N\mathfrak{p}=$

$Q_{p}$

(iv)

$p$

$(O_{\mathfrak{p}} :

\mathfrak{p});ord_{\mathfrak{p}}$

$k_{\mathfrak{p}}.$

$Z_{p}$

$p$

$G_{R}=PSL_{2}(R),$

. If

$G=G_{R}\times G_{\mathfrak{p}}$

$G_{\mathfrak{p}}=PSL_{2}(k_{\mathfrak{p}});PSL_{2}=SL_{2}/\pm 1,$

(subset),

$S\subset G$

$S_{R},$

$S_{\mathfrak{p}}$

$PL_{2}=GL_{2}/$ center;

are set-theoretical projections of to $S$

$G_{R},$

$G_{\mathfrak{p}},$

respectively. (v)

$\mathfrak{H}$

: the complex upper halfplane,

$G_{R}=Aut(\mathfrak{H})$

by Chapter 1, \S 3.

are fields, then Aut $Aut_{k}K$ are the automorphism groups of $K$, or the is its restriction to automorphism groups of $K$ over , respectively. If Aut $K-\{O\}.$ is the multiplicative group

(vi) If $K\supset k$

$K,$

$k$

$k.$

(vii) If

$\sigma\in$

$K,$

$\sigma|_{k}$

$K^{\times}$

$L$

is a

$G_{\mathfrak{p}}-fi$

eld over

then is the set of all non-trivial non-equivalent discrete considered as a complex manifold (see Chapter 2).

$C$

valuations of over $C$ , $L$

,

$\Sigma$

, then is a group, then 1 or denotes the identity element of ; if Int is the inner automorphism is the -conjugacy class of , then containing . If is the Hecke ring with respect to the left -coset decomposition of (if it is defined).

(viii) If

$ I\in\Gamma$

$\Gamma$

$\Gamma$

$ x\mapsto\gamma^{-1}x\gamma$

$(\gamma)$

$\Gamma^{0}\subset\Gamma$

$\Gamma;\{\gamma\}_{\Gamma}$

$\mathcal{H}(\Gamma, \Gamma^{0})$

$\gamma$

$\Gamma^{0}$

(ix) Finally, if

$\Gamma$

$S$

is a set,

$|S|$

or $\#(S)$ denotes its cardinality.

xvii

$\Gamma$

$\gamma\in\Gamma$

CHAPTER 1

Part 1. The group

$\Gamma$

and its -function. $\zeta$

In Part 1 of this chapter, we shall define the -function $\zeta$

$\zeta_{\Gamma}(u)=\prod_{P}(1-u^{\deg P})^{-1}$

of , and prove that $\Gamma$

$\zeta_{\Gamma}(u)=\frac{\prod_{i=1}^{g}(1-\pi_{i}u)(1-\pi_{i}’u)}{(1-u)(1-q^{2}u)}\times(1-u)^{(q-1)(g-1)}$

;

(20) $q=N\mathfrak{p}, g\geq 2, \pi_{i}\pi_{i}’=q^{2}(1\leq i\leq g)$

is compact and is torsion-free. We shall also prove the inequality; , by applying Lemma 10 (M.Kuga), \S 21. These results, particularly the existence of the factor $(1-u)^{(q-1)(g-1)}$ , give a starting point of our problems described in , a consethe introduction. Our formula (20) is, modulo some group theory of space of certain quence of Eichler-Selberg trace formula for the Hecke operators in the automorphic foms of weight 2. However, the proof, starting at Eichler-Selberg formula and ending at (20), is by no means simple, mainly because we do not have a simple proof of Lemma 3 \S 13 Finally, we point out that there is also a diﬀerence in the standpoint; Eichler-Selberg’s left side of the formula comes to the right side of ours; (20). For us, the subject is the set of“elliptic -conjugacy classes”, and not the Hecke operator. We shall begin with the definition of the group

holds, if $q^{2},$

$\pi_{i},$

$|\pi_{i}|,$

$\Gamma$

$ G/\Gamma$

$\pi_{i}’\neq 1,$

$|t_{i}|\leq$

$q^{2}$

$PL_{2}(k_{\mathfrak{p}})$

$($

$)$

$\Gamma$

$\Gamma.$

Discrete subgroup

$\Gamma.$

\S 1. Let (1)

$G=PSL_{2}(R)\times PSL_{2}(k_{\mathfrak{p}})$

resp. be considered as a topological group, and for each subset of , we denote by the set-theoretical projections of to $R$-component (i.e. the first component) resp. $S$

$G$

$S$

$S_{\mathfrak{p}}$

lWe can also prove (20) (for

$\dagger$ $\mathfrak{p}$

2) by using the spectral decomposition of $L^{2}(G/\Gamma)$ .

1

$S_{R}$

2 $k_{\mathfrak{p}}$

-component (i.e. the second component) of . In particular, we have $G$

(2)

$G_{R}=PSL_{2}(R) , G_{\mathfrak{p}}=PSL_{2}(k_{\mathfrak{p}})$

and for any element of $G,$ of ; $x$

$x_{R}$

resp.

$x_{\mathfrak{p}}$

,

denote the $R$-component resp. the

$k_{\mathfrak{p}}$

-component

$x$

(3)

$x=x_{R}\times x_{\mathfrak{p}}.$

, for which subject of our study is a discrete subgroup of \S 2. and are dense in and respectively. So, throughout the following, will always denote such a discrete subgroup of $n_{e}$

$\Gamma$

$G_{R}$

$\Gamma_{\mathfrak{p}}$

$G=G_{R}\times G_{\mathfrak{p}}$

$\Gamma_{R}$

$\Gamma$

$G_{\mathfrak{p}}$

$G.$

EXAMPLE. Let be a prime number, and let denominators are powers of ;

$Z^{(p)}$

$p$

be the ring ofrational numbers whose

$p$

(4)

$Z^{(p)}=\{a/p^{n}|a,n\in Z\}.$

Put (5)

$\Gamma=PSL_{2}(Z^{(p)})=SL_{2}(Z^{p)})/\pm I.$

be the -adic number field. Then, by the injections , the group can be regarded as a subgroup of $G=PSL_{2}(R)\times PSL_{2}(Q_{p})$ . It is discrete in $G$ , since , and if is contained in $PSL_{2}(Z_{p})$ ( : the rin of -adic integers), if

Let

$Q_{p}$

$Z^{Cp)}\rightarrow R,$

$p$

$\rightarrow Q_{p}$

$\Gamma$

$\gamma=\gamma_{R}\times\gamma_{p}\in\Gamma$

$Z_{p}$

$\gamma_{p}$

$g$

$p$

which is a neighborhood of the identity of $PSL_{2}(Q_{p})$ , then is contained in $PSL_{2}(Z)$ , which is discrete in $PSL_{2}(R)$ . It is a simple exercise, in arithmetic ofalgebraic groups, to check that are dense in respectively. Now, for this particular , the projection maps are injective, and the quotient has afinite invariant volume. The former is true in general, as the followming proposition shows; as for the latter, we do not know whether it is true in general, but, curious as it may look, we think that it is quite possible. $\gamma_{R}$

$\Gamma_{R},$

$G_{R},G_{p}$

$\Gamma_{p}$

$\Gamma$

$\Gamma\rightarrow\Gamma_{R},$

$\rightarrow\Gamma_{\mathfrak{p}}$

$ G/\Gamma$

PROPOSITION 1. Let be a discrete subgroup of$G$, for which respectively. Then the projection maps are injective. $\Gamma$

$\Gamma_{R},$

$\Gamma\rightarrow\Gamma_{R},$

PROOF. Let

$\Delta$

be the kemel of the projection

(6)

$\Gamma_{\mathfrak{p}}$

are dense in

$G_{R},$

$G_{\mathfrak{p}}$

$\rightarrow\Gamma_{\mathfrak{p}}$

$\Gamma\rightarrow\Gamma_{R}.$

$\Delta=\{\gamma=\gamma_{R}\times\gamma_{\mathfrak{p}}\in\Gamma|\gamma_{R}=1\}.$

; hence nonnal in , the closure of . So, is a discrete normal subgroup of . On the other hand. it is well-known that if $K$ is any infimite field, then the group $PSL_{2}(K)=SL_{2}(K)/\pm 1$ is simple (as an abstract group). So, ; hence is simple, and hence . The injectivity of follows exactly in the same manner, by using the simplicity of

So

$\Delta_{\mathfrak{p}}\cong\Delta$

is discrete in

$G_{\mathfrak{p}}$

, and normal in

$G_{\mathfrak{p}}$

$\Gamma_{\mathfrak{p}}$

$\Gamma_{\mathfrak{p}}$

$G_{\mathfrak{p}}$

$\Delta_{\mathfrak{p}}$

$G_{\mathfrak{p}}$

$\Delta_{\mathfrak{p}}=\{\eta$

$\Delta=\{I\}$

$\Gamma\rightarrow\Gamma_{\mathfrak{p}}$

$G_{R}.$

So, we can identify the three canonically isomorphic groups: $\Gamma_{R}\cong\Gamma\cong\Gamma_{\mathfrak{p}}.$

$\square $

CHAPTER 1.

3

1. THE GROUP AND ITS -FUNCTION. $\Gamma$

$\zeta$

PROPOSITION 2. Let be a subgroup of $G$ such that the projection maps , are dense in are injective, and that respectively. Let be an open compact subgroup of , and let be the projection to $R$-component of . Then, $G$ (i) is discrete in if and only if is discrete in . Moreover, if (i) is satisfied, then, (ii) the quotient is compact (resp. has afinite invariant volume) if and only if is compact (resp. has afinite invariant volume). $\Gamma$

$\Gamma_{R},$

$\Gamma\rightarrow\Gamma_{R},$

$G_{R},$

$\Gamma_{\mathfrak{p}}$

$U_{\mathfrak{p}}$

$G_{\mathfrak{p}}$

$\Gamma^{0}=\Gamma\cap(G_{R}\times U_{\mathfrak{p}})$

$\Gamma_{R}^{0}$

$G_{\mathfrak{p}}$

$\rightarrow\Gamma_{\mathfrak{p}}$

$\Gamma$

$G_{R}$

$\Gamma_{R}^{0}$

$ G/\Gamma$

$G_{R}/\Gamma_{R}^{0}$

PROOR The first assertion (i) is immediate. The is open, and part is because the “only if‘ part is because is compact (resp. has a is compact. As for (ii), if finite invariant volume), then, there is a subspace of which is compact (resp. has a , it follows finite invariant volume) such that . Since we have ; which proves the only if’ part. Conversely, if immediately that is non-compact (resp. has an infinite volume), then there is an open subset of such that the restriction to of the natural map is injective, and that : is non-compact (resp. has an arbitrarily large volume). Put . Then, the $F$ $F$ restriction to ofthe natural map : is injective, and is non-compact (resp. has an arbitrarily large volume); which proves the if’ part of (ii). $iF$

’

$U_{\mathfrak{p}}$

$G_{R}/\Gamma_{R}^{0}$

$U_{\mathfrak{p}}$

$G_{R}$

$K_{R}$

$G_{R}=K_{R}\cdot\Gamma_{R}^{0}$

$G_{\mathfrak{p}}=U_{\mathfrak{p}}\cdot\Gamma_{p}$

$ G=(K_{R}\times U_{p})\cdot\Gamma$

$G_{R}/\Gamma_{R}^{0}$

$F_{R}$

$F_{R}$

$\varphi_{R}$

$G_{R}$

$F_{R}$

$G_{R}\rightarrow G_{R}/\Gamma_{R}^{0}$

$F=F_{R}\times U_{\mathfrak{p}}$

$\varphi$

$ G\rightarrow G/\Gamma$

$\square $

\S 3. Now, $G_{R}=PSL_{2}(R)$ acts on the complex upper halfplane $\mathfrak{H}=\{z\in C|{\rm Im} z>0\}$

as: (7)

$G_{R}\ni g_{R}=\left(\begin{array}{ll}a & b\\c & d\end{array}\right)$

:

$\mathfrak{H}\ni Z\mapsto g_{R}\cdot z=\frac{az+b}{cz+d}\in \mathfrak{H}.$

As is well-known, acts transitively on , and is identﬃed with the group of all automorphisms of the complex Riemann surface . Since is identﬃed with its projection also acts on . Two points will be called equivalent (or, more precisely, -equivalent), if there is an element such that . We note $A$ point that, since is dense in , each equivalence class is also dense on , we put will be called a -fixed point, if its stabilizer in is infinite. For each $G_{R}$

$Aut(\mathfrak{H})$

$\mathfrak{H}$

$\Gamma$

$\mathfrak{H}$

$\Gamma_{R}\subset G_{R},$

$\Gamma$

$z,z’\in \mathfrak{H}$

$\mathfrak{H}$

$\Gamma$

$G_{R}$

$\Gamma_{R}$

(8)

$\mathfrak{H}.$

$\Gamma$

$z\in \mathfrak{H}$

$\left\{\begin{array}{l}G_{z,R} = \{g_{R}\in G_{R}|g_{R}\cdot z=z\}\cong R/Z\\\Gamma_{z,R} =\Gamma_{R}\cap G_{z,R}=\{\gamma_{R}\in\Gamma_{R}|\gamma_{R}\cdot z=z\}.\end{array}\right.$

be the subgroup of isomorphism:

Let

$\Gamma_{R}\cdot z$

$\Gamma$

$z\in \mathfrak{H}$

$\gamma_{R}\cdot z=z’$

$\gamma\in\Gamma$

$\Gamma_{z},$

$\Gamma_{z,\mathfrak{p}}$

$\Gamma,$

$\Gamma_{\mathfrak{p}}$

respectively which correspond to

$\Gamma_{z,R}$

by the canonical

$\Gamma_{z,R}\cong\Gamma_{z}\cong\Gamma_{z,\mathfrak{p}}.$

(9)

So,

$\left\{\begin{array}{ll}\Gamma_{z} & = \{\gamma\in\Gamma|\gamma_{R}\cdot z=z\}\\\Gamma_{z,\mathfrak{p}} & = \{\gamma_{\mathfrak{p}}\in\Gamma_{\mathfrak{p}}|\gamma\in\Gamma_{z}\}=\{\gamma_{\mathfrak{p}}\in\Gamma_{\mathfrak{p}}|\gamma_{R}\cdot z=z\}.\end{array}\right.$

$z\in \mathfrak{H}$

is a -fixed point if and only if $\Gamma$

(10)

$\Gamma_{z,R}\cong\Gamma_{z}\cong\Gamma_{z,\mathfrak{p}}$

are infinite. Let

$\wp(\Gamma)$

be the set of all -equivalence classes of all -fixed points on . We shall see later, that is analogous, in various sense, to the set ofall prime divisors of an algebraic function field of one variable over the finite field , where An element will be called elliptic if it has a fixed point on . So, is elliptic if and only if has imaginary eigenvalues, and hence if and only if $|trg_{R}|0.\end{array}\right.$

$br$

So, by Proposition 5 and (33), we get $A_{m}=N_{m}+(q-1)\sum_{k=1}^{m-1}q^{k-1}N_{m-k},$

which settles (36). Now, (37) is a formal consequence of (36). In fact, it can be checked directly, by substituting (36) on the right side of (37). $\square $

CHAPTER 1.

1,

13

THE GROUP AND ITS -FUNCTION. $\Gamma$

$\zeta$

The proof of Theorem 1 assuming Lemmas 2, 3.

\S 14. We have (37)

$N_{m}=A_{m}-(q-1)\sum_{k=\iota}^{m-1}A_{m-k}.$

with Apply Lemma 1 for . Since we can identify consider as representations of . Since, by (30), we have $q^{2m}+q^{2m-1}$ , we get $\mathcal{H}(r,r^{0})$

$\tilde{\Delta}=\Gamma_{R},\Delta=\Gamma_{R}^{0}$

(38)

$A_{m}=q^{2m}+q^{2m-1}-tr\rho(\Gamma^{m})$

$(m\geqq 1)$

.

By substituting (38) in (37), we get (39)

$N_{m}=q^{2m}+q-tr\rho\{\Gamma^{m}-(q-1)\sum_{k=1}^{m-1}\Gamma^{m-k}\}.$

Since tr $\rho(I)=g$ , the genus of (40)

$r_{R}^{0}\backslash \mathfrak{H}$

,

we get

$N_{m}=q^{2m}+1-(q-1)(g-1)-tr\rho\{\Gamma^{m}-(q-1)\sum_{k=1}^{m}\Gamma^{m-k}\}.$

On the other hand, by (31) (Lemma 2’) we get (41)

$\frac{1-qu}{1-u}\sum_{m=0}^{\infty}\Gamma^{m}u^{m}=\frac{1-q^{2}u^{2}}{1-(\Gamma^{1}-q+1)u+q^{2}u^{2}}$

;

and by a simple computation, we see that the left side of (41) is equal to (42)

$\sum_{m=1}^{\infty}\{\Gamma^{m}-(q-1)\sum_{k=1}^{m}\Gamma^{m-k}\}u^{m}.$

Put (43)

$1-(\Gamma^{1}-q+1)u+q^{2}u^{2}=(1-\pi u)(1-\pi’u)$

formally, with $\pi/=t\pi=q^{2}$ . Then, $\frac{1}{(1-\pi u)(1-\pi’u)}=\sum_{m=0}^{\infty}(\pi^{m}+\pi^{m-1}\pi’+\cdots+\pi^{\prime m})u^{m}$

(44) $=1+\sum_{m=1}^{\infty}(\pi^{m}+\pi^{\prime m})u^{m}+q^{2}u^{2}\frac{1}{(1-\pi u)(1-\pi’u)}$

hence we get (45)

$\frac{1-q^{2}u^{2}}{(1-\pi u)(1-\pi’u)}=1+\sum_{m=1}^{\infty}(\pi^{m}+\pi^{\prime m})u^{m}.$

Therefore, by (41), we get (46)

,

we

$d(\Gamma^{m})=|r^{\triangleleft}\backslash \Gamma^{m}|=$

$\mathcal{H}(\Gamma,\Gamma^{\triangleleft})$

$ d,\rho$

$\mathcal{H}(\Gamma_{R},\Gamma_{R}^{0})$

$\Gamma^{m}-(q-1)\sum_{k=1}^{m}\Gamma^{m-k}=\pi^{m}+\pi^{\prime m} (m\geq 1)$

.

;

14

This is a formal computation, but this shows that if is a linear representation of the , and if we put ring $\chi$

$\mathcal{H}(\Gamma,\Gamma^{0})$

$1-\alpha(\Gamma^{1})-q+1)u+q^{2}u^{2}=(1-\pi u)(1-\pi’u)$ ,

then (47)

$\chi(\Gamma^{m})-(q-1)\sum_{k=1}^{m}\chi(\Gamma^{m-k})=\pi^{m}+\pi^{\prime m} (m\geq 1)$

holds. Now, by Lemma 1, is a direct sum of linear representations: $g$

$\rho$

$\rho=\chi_{1}\oplus\cdots\oplus\chi_{g}$

;

so, by putting (48)

$1-(\chi_{i}(\Gamma^{1})-q+1)u+q^{2}u^{2}=(1-\pi_{l}u)(1-\pi_{i}’u) (1\leq i\leq g,\pi_{i}\pi_{i}’=q^{2})$

we get (49)

$\chi_{i}(\Gamma^{m})-(q-1)\sum_{k=1}^{m}\chi_{i}(\Gamma^{m-k})=\pi_{i}^{m}+\pi_{i}^{\prime m} (1\leq i\leq g, m\geq 1)$

.

So, by summing over $i(1\leq i\leq g)$ , we obtain: (50)

$tr\rho\{\Gamma^{m}-(q-1)\sum_{k=1}^{m}\Gamma^{m-k}\}=\sum_{i=1}^{g}(\theta_{i}+\pi_{i}^{\prime m});$

and hence, by (40), we get (51)

$N_{m}=q^{2m}+1-(q-1)(g-1)-\sum_{i=1}^{g}(\pi_{i}^{m}+\pi_{i}^{\prime m}) (m\geq 1)$

;

and hence we get (52)

$\zeta_{\Gamma}(u)=\exp\sum_{m=1}^{\infty}\frac{N_{m}}{m}u^{m}=\frac{\prod_{i=1}^{g}(1-\pi_{i}u)(1-t_{i}u)}{(1-u)(1-q^{2}u)}\times(1-u)^{(q-1Xg-1)}.$

Since (48) are the eigenvalues of $1-\phi(\Gamma^{1})-q+1)u+q^{2}u^{2}$ , we have (53)

$\zeta_{\Gamma}(u)=\frac{\det\{1-\phi(\Gamma^{1})-q+1)u+q^{2}u^{2}\}}{(1-u)(1-q^{2}u)}\times(1-u)^{(q-1)(g-1)}.$

That $\pi_{i},\#_{i}(1\leq i\leq g)$ are algebraic integers follows immediately from (51) (for $1,$

$\cdots,2g)$

$m=$

.

$\square $

So, we have also shown:

ASUPPLEMENT TO THEOREM 1. The numerator of the main factor of (54)

$\zeta_{\Gamma}(u)$

$\prod_{i=1}^{g}(1-\pi_{l}u)(l-\pi_{i}’u)=\det\{1-(\rho(\Gamma^{1})-q+1)u+q^{2}u^{2}\}.$

is given by:

CHAPTER 1.

1. THE GROUP AND ITS -FUNCTION.

15

$\Gamma$

$\zeta$

Proofs of Lemmas 2, 3.

\S 15. Put (55)

$X=PL_{2}(k_{\mathfrak{p}})=GL_{2}(k_{\mathfrak{p}})/k_{\mathfrak{p}}^{x}.$

Then, for any element

$x\in X$,

are all contained in

we can take its representative are in . Put

$\left(\begin{array}{ll}a & b\\c & d\end{array}\right)mod k_{\mathfrak{p}}^{\times}$

such that

. Then, is a non-negative integer, well-defined by . We shall call it the length of . It is clear that we have $a,$

$b,c,d$

$O_{\mathfrak{p}}$

, but not all

$\mathfrak{p}$

$(ad-bc)O_{\mathfrak{p}}=\mathfrak{p}^{l(x)}$

$x$

(56)

$l(x)$

$x$

$l(x_{1}x_{2}\cdots x_{n})\leq l(x_{1})+\cdots+l(x_{n})$

$\equiv l(x_{1})+\cdots+l(x_{n}) (mod 2)$ ,

for any

$x_{1},$

$\cdots,x_{n}\in X$

Put

(57)

$X_{l}=\{x\in X|l(x)=l\}.$

In particular, (58)

$X_{0}=PL_{2}(O_{\mathfrak{p}})=GL_{2}(O_{\mathfrak{p}})/\mathcal{U}_{\mathfrak{p}}$

is an open compact subgroup of , and it is well-known by elementary divisor theory, that each consists of a single -double-coset; $X^{s}$

$X_{l}$

(59)

$X_{0}$

$X_{l}=X_{0}(_{0}^{p^{l}}01\rangle X_{0}$

,

where

$p$

is any prime element of

$k_{\mathfrak{p}}.$

Since is open compact, for any $x\in X$, the subgroups and are commensurable with each other; hence for each is finite, and the Hecke ring can be defined. Moreover, since $l(x^{-1})=l(x)$ for each $x\in X$, each is self-inverse, and hence is commutative. Now, the following lemma is a very well-known one: $X_{0}$

$x^{-1}X_{0}x$

$X_{0}$

$l\geq 0$

$|X_{0}\backslash X_{l}|$

$\mathcal{H}(X,X_{0})$

$X_{l}$

$\mathcal{H}(X,X_{0})$

LEMMA 4. Let be a prime element of , and let matrices mod kd foms a set of representatives $p$

$k_{\mathfrak{p}}$

$l\geq 1$

. Then the following set of

$ofX_{0}\backslash X_{l\prime}.$

$m,n\geq 0,m+n=l$

(60)

$\{\left(\begin{array}{ll}p^{m} & \alpha\\ 0 & p^{n}\end{array}\right);$

$\alpha.$

representatives of

$O_{\mathfrak{p}}(mod \mathfrak{p}^{n})$ 0(mod \mathfrak{p})\}.$ $n\alpha\not\equiv

If $m,n$ are both $>0,$ the In particular, we have (61)

hence we have

$X_{1}=X_{0}\left(\begin{array}{ll}p & 0\\0 & 1\end{array}\right)+\sum_{\alpha mod \mathfrak{p}}X_{0}(_{0}^{1}p\alpha)$

$|X_{0}\backslash X_{1}|=1+q.$

(disjoint);

16

\S 16. Now we shall prove the following two equivalent lemmas; Lemmas 5, 5’. (disjoint). then, LEMMA 5. Put (i) For each $i(0\leq i\leq q)$, there exists a unique $suffi\kappa j(0\leq j\leq q)$ such that We shall put $j=\rho(\iota)(0\leq i\leq q)$ . (ii) Any element $x\in X_{l}(l\geq 0)$ can be expressed uniquely in the form: $X_{1}=\sum_{i=0}^{q}X_{0}\pi_{i}$

(62)

$x=u\pi_{i_{1}}\pi_{i_{2}}\cdots\pi_{i_{l}}$

with

,

$u\in X_{0},$

$i_{n}\neq\rho(i_{n+1})(1\leq\forall n\leq l-1)$

Conversely, an element $x\in X$ of the $fom(62)$ is contained in (63)

.

. In short, we have

$X_{l}=\sum’X_{0}\pi_{i_{1}}\cdots\pi_{i_{l}},$

where the disjoint union $l)$

$X_{l}$

$\pi_{j}\pi_{i}\in X_{0}.$

$\sum’$

is over all

$\{i_{1}, \cdots, i_{l}\}$

such that

$i_{n}\neq\rho(i_{n+1})$

for all

$ n(1\leq n\leq$

.

We note that (i) is trivial, since $j=\rho(i)$ is uniquely determined by is merely for a better understanding of (ii).

LEMMA 5’. As elements of (64)

$\mathcal{H}(X,X_{0})$

,

$X_{0}\pi_{j}=X_{0}\pi_{i}^{-1}$

. This

we have

$X_{1}^{2}=X_{2}+(q+1)X_{0},$

(65)

$X_{1}X_{l}=X_{l}X_{1}=X_{l+1}+qX_{l-1} (l\geq 2)$

.

This Lemma 5’ is more or less well-known. We shall prove Lemma 5 (ii) and Lemma 5’ in the followming order; , Lemma 5(ii) for any Lemma Lemma 5 (ii) for a particular $ 5’\Rightarrow$

$\pi_{0},$

PROOR Let tatives of

$p$

$\pi_{0},$

$\cdots,\pi_{q}\Rightarrow$

be a prime element of , and let . Put $k_{\mathfrak{p}}$

$\alpha_{1}=0,\alpha_{2},$

,

$\cdots$

$\alpha_{q}$

$\cdots$

$\pi_{q}.$

be a set ofrepresen-

$O_{\mathfrak{p}}mod \mathfrak{p}$

(66)

$\pi_{0}=(_{0}^{p}01\rangle,\pi_{i}=(_{0}^{1}\alpha_{i\rangle}p (1\leq i\leq q)$

By (61), we have

$X_{1}=\Sigma_{i=0}^{q}X_{0}\pi_{i}$

(disjoint). Since

$\pi_{0}\pi_{i}=\left(\begin{array}{ll}p & p\alpha_{i}\\0 & p\end{array}\right)\equiv\left(\begin{array}{ll}1 & \alpha_{i}\\0 & l\end{array}\right) (mod k_{p}^{\times})$

we have

$\pi_{0}\pi_{i}\in X_{0}$

for

$1\leq i\leq q$

$\pi_{1}\pi_{0}\epsilon X_{0}$

(67)

,

, and hence $\rho(\iota)=0(1\leq i\leq q)$ . Since

$\pi_{1}\pi_{0}=\left(\begin{array}{ll}p & 0\\0 & p\end{array}\right)\equiv(_{0}^{1}01\rangle (mod k_{\mathfrak{p}}^{\times})$

we have

.

,

; hence $\rho(0)=1$ . So, to show Lemma 5 (ii), it is enough to show that: $X_{l}=\sum_{s=0}^{l}\sum_{i_{1}\cdot\cdots.i_{l-s}\geq 1t_{l-s^{>1ifs>0}}}X_{0}\pi_{i_{1}}\cdots\pi_{i_{l-s}}\pi_{0}^{s}$

(disjoint).

But we have $\pi_{i_{1}}\cdots\pi_{i_{l-s}}\pi_{0}^{s}=\left(\begin{array}{llll}p^{s} & \alpha_{t_{l-s}}+\alpha_{i_{l-\prime-1}}p+ & \cdots & +\alpha_{i_{1}}p^{l-s-l}\\0 & p^{l-s} & & \end{array}\right).$

CHAPTER 1.

17

1. THE GROUP AND ITS -FUNCTION. $\Gamma$

$\zeta$

Hence, (67) follows immediately from Lemma 4. So, Lemma 5 (ii) is proved for the

. given by (66). This also shows particular be as in (66). Then we have Now let us prove Lemma 5’. Let ; hence , multiplicity being taken into account. Hence $\pi_{0},$

$|X_{0}\backslash X_{l}|=q^{l}+q^{l-1}(l\geq 1)$

$\cdots,\pi_{q}$

$\pi_{0},$

$\cdots,\pi_{q}$

$X_{1}=$

$X_{1}^{2}=\sum_{i,j}X_{0}\pi_{j}\pi_{i}$

$\sum_{i=0}^{q}X_{0}\pi_{i}$

$X_{1}^{2}=\sum_{i,j j\neq\rho(\iota)}X_{0}\pi_{j}\pi_{i}+\sum_{i,j j=\rho(\iota)}X_{0}\pi_{j}\pi_{i}=X_{2}+\sum_{j=\rho(\iota)}X_{0}=X_{2}+(q+1)X_{0}.$

By Lemma 5 (ii) for these

$\pi_{0},$

$\cdots,\pi_{q}$

,

we have

$X_{l}=\sum_{i_{n}\neq\rho(i_{n+1}),\forall n}X_{0}\pi_{i_{1}}\cdots\pi_{i_{l}}$

. So,

$X_{1}X_{l}=\sum_{i=0}^{q}\sum_{i_{n}\neq\rho(i_{n+1})}X_{0}\pi_{i}\pi_{i_{1}}\cdots\pi_{i_{l}}$

$=\sum_{n_{1}+1}X_{0}\pi_{i}\pi_{i_{1}}\cdots\pi_{i_{l}}+\sum_{)t_{n}\neq\rho(i),i_{n}\neq\rho(i_{n_{1}+1} ,i\neq\rho(i)j=\rho(i)}.X_{0}\pi_{i}\pi_{i_{1}}\cdots\pi_{i_{l}}$

$=X_{l+1}+\sum_{nn\rho_{\hslash}+1 ,1\leq\leq l-1}X_{0}\pi_{i_{2}}\cdots\pi_{i_{l}}i\neq(i).$

$=X_{l+1}+\sum_{i_{1\neq\rho(i_{2})}}\sum_{t_{n}\neq\rho(i_{n+1}),2\leq n\leq l-1}X_{0}\pi_{i_{2}}\cdots\pi_{i_{l}}$

$=X_{l+1}+qX_{l-1} (\geq 2)$

.

Since is commutative, we have $X_{l}X_{1}=X_{1}X_{l}=X_{l+1}+qX_{l-1}$ ; hence Lemma 5’ is proved. of representatives Finally, let us prove Lemma 5 (ii) for an arbitrary set of . By (65), we obtain $\mathcal{H}(X,X_{0})$

$\pi_{0},\pi_{1},$

$\cdots,\pi_{q}$

$X_{0}\backslash X_{1};X_{1}=\sum_{i=0}^{q}X_{0}\pi_{i}$

(68)

$X_{1}^{l}=X_{l}+cX_{l-2}+c’X_{l-4}+\cdots (l\geq 1)$ ,

where $c,c’,$ are non-negative integers. In fact, it is trivial for $l=1$ ; so, assume that (68) , and multiply is true for some on both sides. Then ﬀom (65) follows directly ﬃat (68) is also true for $l+1$ . Now, the expression of by the formal sum of $1eftX_{0}$ -cosets, multiplicities being taken into account, will be $\cdots$

$l\geq 1$

$X_{1}$

$X_{1}$

(69)

$\sum X_{0}\pi_{i_{1}}\cdots\pi_{i_{l}}=\sum’X_{0}\pi_{i_{1}}\cdots\pi_{i_{l}}+$

lower length terms,

, is , and the second one, where the first formal is over all , with for all $n(1\leq n\leq l-1)$ . On the other hand, over all . Thus, by , which is equal to in (69) is the number of terms under comparing (68) and (69), we see that all left cosets under in (69) must be mutually distinct, elements of such left cosets have length , and that $0\leq i_{1},$ $\cdots,$ $i_{l}\leq q$

$s\iota m\sum$

$0\leq i_{1},$ $\cdots,i_{l}\leq q$

$\sum’$

$i_{n}\neq\rho(i_{n+1})$

$\sum’$

$q^{l}+q^{l-1}$

$|X_{0}\backslash X_{l}|$

$\sum’$

$X_{0}$

$l$

$X_{0}$

$X_{l}=\sum’X_{0}\pi_{i_{1}}\cdots\pi_{i_{l}}$

(disjoint);

which proves Lemma 5 (ii). COROLLARy

$\square $

1. We have

(70)

REMARK. Since

$|X_{0}\backslash X_{l}|=|X_{l}/X_{0}|=q^{l}+q^{l-1}$

$X_{t}^{-1}=X_{t}$

,

we have

for

$|X_{l}/X_{0}|=|X_{0}\backslash X_{l}|.$

$l\geq 1.$

18

COROLLARY 2. We have (71)

$\sum_{l\overline{-}0}^{\infty}X_{l}u^{l}=\frac{1-u^{2}}{1-X_{1}u+qu^{2}},$

as an identity between two fomal power series of with coeﬃcients in . PROOF. That follows directly from Lemma 5’. $\mathcal{H}(X,X_{0})$

$u$

$(1-X_{1}u+qu^{2})\sum_{l-0}^{\infty}-X_{l}u^{l}=1-u^{2}$

$\square $

\S 17. The proof of Lemma 2. Put $\chi=\{x\in X|l(x)\equiv 0(mod 2)\}$

(72)

$=\bigcup_{l=0}^{\infty}X_{2l}.

$

forms a subgroup of $X$ with index2. It is easy to see that if $ X\ni x\mapsto$ is ﬃe homomorphism of $X$ onto induced from the determinant , then we map: have Then,

$\det x\in k_{\mathfrak{p}}^{x}/k_{\mathfrak{p}}^{\times 2}$

$k_{\mathfrak{p}}^{x}/k_{\mathfrak{p}}^{\times 2}$

$GL_{2}(k_{\mathfrak{p}})\ni x\mapsto\det x\in k_{\mathfrak{p}}^{x}$

$X=\{x\in X|\det x\in\epsilon^{2}\mathcal{U}_{\mathfrak{p}}/k_{p}^{x2}\}$

(73)

$=\{x\in X|ord_{\mathfrak{p}}(\det x)\equiv 0 (mod 2)\}$ $=PL_{2}(O_{\mathfrak{p}})\cdot PSL_{2}(k_{\mathfrak{p}})=X_{0}\cdot G_{\mathfrak{p}}.$

On the other hand, (71) gives rise to $2 \sum_{l=0}^{\infty}X_{2l}u^{2l}=\sum_{l\overline{-}0}^{\infty}X_{l}u^{l}+\sum_{l=0}^{\infty}X_{l}(-u)^{l}=\frac{2(1-u^{2})(1+qu^{2})}{(1+qu^{2})^{2}-X_{1}^{2}u^{2}}$

;

hence we get (74)

$\sum_{l=0}^{\infty}X_{2l}u^{l}=\frac{(1-u)(1+qu)}{1-(X_{2}-q+1)u+q^{2}u^{2}}.$

So, to prove Lemma 2, it is enough to show that

$\mathcal{H}(G_{\mathfrak{p}}, U_{\mathfrak{p}})$

and

$\mathcal{H}(X,X_{0})$

are canonically . To see this,

isomorphic, i.e. there is an isomorphism which maps on we remark that, in general, if are three groups such that $X_{2l}(l\geq 1)$

$Y_{l}$

; , $x^{-1}H_{2}x\sim H_{2}(\forall x\in H_{1};H_{2}=H_{1}\cap G_{2})$ , and commensurability, , then the two Hecke rings that defined with respect to (say) left coset decompositions are canonically isomorphic; i.e., corresponds to . This follows immediately from the definition of the Hecke rings. Thus, to show that and are X_{2l}(l\geq 0)$ $Y_{l}\mapsto enoug , it is isomorphic by , since we to check $G_{1}\supset G_{2},$

$x^{-1}G_{2}x\sim G_{2}$

$H_{1}$

$G_{1}=G_{2}H_{1}$

$\forall x\in G_{1})$

$(\sim:$

$G_{2}h_{1}G_{2}\cap H_{1}=H_{2}h_{1}H_{2}(\forall h_{1}\in H_{1})$

$H_{2}h_{1}H_{2}\in \mathcal{H}(H_{1},H_{2})$

$\mathcal{H}(G_{1},G_{2}),$ $\mathcal{H}(H{}_{1}H_{2})$

$G_{2}h_{1}G_{2}\in \mathcal{H}(G_{1},G_{2})$

$\mathcal{H}(G_{\mathfrak{p}}, U_{\mathfrak{p}})$

$\mathcal{H}(X,X_{0})$

$X_{2l}\cap G_{\mathfrak{p}}=Y_{l}(l\geq 0)$

know that

$Y_{l}$

is a single

$U_{\mathfrak{p}}$

double coset. But

$Y_{l}=U_{\mathfrak{p}}(_{0}^{p^{l}}p^{-l\rangle U_{\mathfrak{p}}}0$

with elementary divisors hence the Lemma 2 is proved. $g_{\mathfrak{p}}\in G_{\mathfrak{p}}=PSL_{2}(k_{\mathfrak{p}})$

$p^{-l},p^{l}$

consists ofall elements

; i.e., all elements

$g_{\mathfrak{p}}\in G_{\mathfrak{p}}\cap X_{2l}$

;

$\square $

CHAPTER 1.

19

1. THE GROUP AND ITS -FUNCTION. $\Gamma$

$\zeta$

\S 18. For the proof of Lemma 3, we need some more lemmas, which are direct consequences of Lemma , Let . We shall say that the product is free, if $5^{6}$

$x_{1},$

$x_{n}\in X=PL_{2}(k_{\mathfrak{p}})$

$\cdots$

(75)

$x_{1}\cdots\cdot\cdot x_{n}$

$l(x_{1}\cdots\cdot\cdot x_{n})=l(x_{1})+\cdots+l(x_{n})$

holds. LEMMA 6. Let is also free.

$x,$

$y,z\in X,$

$y\not\in X_{0}$

.

If the two pmducts

$x\cdot y,$ $y\cdot z$

are free, then the pmduct

$x\cdot y\cdot z$

PROOR Let

$\pi_{0},$

$xu’=u’’\pi_{v\iota}\cdots\pi_{v_{n}}$

By assumption, Lemma 5,

be as in Lemma 5, and factorize $u’’\in X_{0},$ $l=l(z),$ $m=l(y)>0,n=l(x)$ (see , where Lemma 5). are free products; hence . Therefore, by has length $l+m+n.$

$\cdots,\pi_{q}$

$z=u\pi_{\lambda_{1}}\cdots\pi_{\lambda_{l}},$

$yu=u’\pi_{\mu_{1}}\cdots\pi_{\mu_{m}},$

$u,$ $u’,$

$y\cdot z,x\cdot y$

$\pi_{\mu_{m}}\pi_{\lambda_{1}}\not\in X_{0},$

$\pi_{v_{n}}\pi_{\mu 1}\not\in X_{0}$

$xyz=u’’\pi_{v}\cdots\pi_{v_{n}}\pi_{\mu\iota}\cdots\pi_{\mu_{m}}\pi_{\lambda_{1}}\cdots\pi_{\lambda_{l}}1$

LEMMA 7. Let

of . $xy$

Then,

$x\cdot y$

be a ﬀee product, and let

$x=u\pi_{i_{1}}\cdots\pi_{i_{m}}u^{\prime-1},$ $y=u’\pi_{i_{m+1}}\cdots\pi_{i_{l}}$

$\square $

$xy=u\pi_{i_{1}}\cdots\pi_{i_{l}}$

with some

be the factorization (62) , and with $m=l(x)$ .

$u’\in X_{0}$

PROOR Let be the factorization (62) for . Since the factorization of $xy$ can be obtained by factorizations of and , and then by carrying the elements of to the left (no influence to -side!), we see directly by the uniqueness oﬀactorization (62) for $xy$ that $j_{m+1}=i_{m+1},$ , , nd hence for some $y=u’\pi_{j_{m+1}}\cdots\pi_{j_{l}}$

$y$

$x$

$X_{0}$

$y$

$y$

$\cdots$

$j_{l}=i_{l}$

$u’\in X_{0}.$

$y=u’\pi_{i_{m+1}}\cdots\pi_{i_{l}}$

$\square $

LEMMA 8. Let $x,y\in X$, and put $l(xy)=l(x)+l(y)-2d$ Then $d\leqq l(x),$ $l(y)$ ; and if are free products with $d\leqq l(x’),$ $l(y’)$, then $l(x’y’)=l(x’)+l(y’)-2d$

$x=x’’\cdot x’,$ $y=y’\cdot y’’$

PROOF. The first assertion is clear.7 Let $x=u\pi_{i_{1}}\cdots\pi_{i_{l}}, y=u’\pi_{j_{1}}\cdots\pi_{j_{m}}$

be the factorizations (62) for

$x,$ $y$

. By Lemma 7,

$x’=u’’\pi_{i_{s}}\cdots\pi_{i_{l}}, y’=u’\pi_{j\iota}\cdots\pi_{j_{l}}u’’’$

with

$u’’,$ $u’’’\in X_{0},$

$l(x’)=l-s+1\geq d,$ $l(y’)=t\geq d$.

It is enough to prove that

$l(\pi_{i_{s}}\cdots\pi_{i_{l}}u’\pi_{j_{1}}\cdots\pi_{j_{t}})=(l-s+1)+t-2d.$

This can be seen easily from the process of obtaining the factorization (62) for $xy$ from that of and given above. $x$

$y$

$\square $

LEMMA 9. Let

$x_{1},$

$\cdots,$

$x_{n}$

be any elements $ofX$ andput

$l(x_{i}x_{i+1})=l(x_{i})+l(x_{i+1})-2d_{i} (1\leq i\leq n-1)$

If

$l(x_{i+1})>d_{i}+d_{i+1}$

(76)

holds 8 for all $i(1\leq i\leq n-2)$, then

$l(x_{1}\cdots x_{n})=l(x_{1})+\cdots+l(x_{n})-2(d_{1}+\cdots+d_{n-1})$

6 They are given in Y. Ihara [16]. 7 Since $x=xy\cdot y^{-1}$ , we have $l(x)\leq

8Where we put

.

$d_{n}=0.$

l(xy)+l(y)$

; thus we get $1(xy)\geq|l(x)-l(y)|.$

.

20

with $l(a_{i})=d_{i-1},l(b_{i})=l(x_{i})-$ into free product $d_{i}-d_{i-1}>0,l(c_{i})=d_{i}$ (here we understand $a_{1}=c_{n}=1$ ). Lemma 8 shows that PROOF. Factorize each

$x_{i}=a_{i}b_{i}c_{i}$

$x_{i}$

$ c_{i}a_{i+1}\in$

and that $l(b_{i}c_{i}a_{i+1}b_{i+1})=l(b_{i})+l(b_{i+1})$ , and hence the products are free. Now our , and hence also ﬃe product lemma follows directly from Lemma 6.

$X_{0}(1\leq i\leq n-1)$ ,

$(b_{i}c_{i}a_{i+1})\cdot(b_{i+1}c_{i+1}a_{i+2})$

$(b_{j}c_{i}a_{i+1})\cdot b_{i+1}$

$\square $

COROUARY. Let $x_{1}\cdot x_{2},$

$\cdots,$

$x_{n-1}\cdot x_{n}$

with are allfree, the product $x_{1},$

, $x_{n}\in

$\cdots$

X$

$l(x_{2}),$

$\cdots$

, $l(x_{n-1})>0$ .

Then,

if the products

is also fiee.

$x_{1}\cdots x_{n}$

\S 19. The proof of Lemma 3. Recall the definitions; $\Gamma^{l}=\{\gamma\in\Gamma|\gamma_{\mathfrak{p}}\in Y_{l}=U_{\mathfrak{p}}\left(\begin{array}{ll}p^{l} & 0\\0 & p^{-l}\end{array}\right)U_{\mathfrak{p}}\} (l\geqq 0)$

,

belongs to , we , and is a prime element of . When where . To avoid unnecessary suﬃces, we shall not make distinction between and put as ; and consider as (dense) subgroup of . Also, we consider . We note here, that the definitions of the fimctions $l(x)$ are a subgroup of . We shall use the symbol $l(x)$ and on $X$; in fact, we have diﬀerent on . We shall further put $L(x)=l$ for exclusively in the sense that $l(x)=l$ for $\Gamma^{l}$

$U_{\mathfrak{p}}=PSL_{2}(O_{\mathfrak{p}})$

$\gamma\in\Gamma$

$k_{\mathfrak{p}}$

$p$

$\Gamma$

$l=l(\gamma)$

$\Gamma$

$a$

$\Gamma_{\mathfrak{p}}$

$G_{\mathfrak{p}}=PSL_{2}(k_{\mathfrak{p}})$

$G_{\mathfrak{p}}$

$X=PL_{2}(k_{\mathfrak{p}})$

$Y_{l}=G_{\mathfrak{p}}\cap X_{2l}$

$G_{\mathfrak{p}}$

$x\in X_{l}.$

$x\in Y_{l}$

Thus, we have (77)

for

$l(x)=2L(x)$

of The product holds. We shall show that any element $\gamma_{1}\gamma_{2}\cdots\gamma_{n}$

of elements of

$\Gamma^{1}$

$\gamma_{1},$

$\cdots,$

$\gamma_{n}\in\Gamma$

$x\in G_{\mathfrak{p}}.$

is $called\ell ee$ , if

$l(\gamma_{1}\cdots\gamma_{n})=l(\gamma_{1})+\cdots+l(\gamma_{n})$

with $l(\gamma)=l(l=1,2, \cdots)$ is a free product

$\gamma\in\Gamma$

;

(78)

$\gamma=\gamma_{1}\gamma_{2}\cdots\gamma_{l}$

;

$\cdots,\gamma_{l}\in\Gamma^{1}.$ $\gamma_{1},$

, and prove it for $l(\gamma)=l.$ In fact, it is trivial for $l=1$ . Assume that it is true for $\cdots,x_{2l}\in X_{1}$ . Since with By Lemma 5, we can put . Then we contained in , there is an element , and hence by the induction assumption, we have have ; hence we get wifh be a primitive elliptic -conjugacy class of degree . By Proposition Now let wifh 6, we can assume, without loss of generality, that $l(\gamma)=d$. Put are free; but moreover, the product , . Then the products , which is a must also be hee. In fact, if not, then . Therefore, the products contradiction, since by Lemma 6, we have $l(\gamma^{r})=|r|l(\gamma)=|r|d$ Z. Another for any etc. are also ﬀee, and we have , then, (62) of are the factorizations remark is that, if $l(\gamma)\leqq l-1$

$\gamma=x_{1}x_{2}\cdots x_{2l}$

$\gamma_{l}\in\Gamma$

$PL_{2}(O_{\mathfrak{p}})\cdot G_{\mathfrak{p}}=X$

$l(\gamma_{l})=1,$

$PL_{2}(O_{\mathfrak{p}})x_{2l-1}x_{2l}$

$\gamma\gamma_{l}^{-1}=$

$l(\gamma\gamma_{l}^{-1})=l-1$

$\gamma_{l-1}\in\Gamma^{1}$

$\gamma_{1}\cdots\gamma_{l-1}$

$PL_{2}(O_{p})\Gamma_{\mathfrak{p}}=$

$x_{1},$

$\gamma_{1},$

$\gamma=\gamma_{1}\gamma_{2}\cdots\gamma_{l}.$

$\cdots,$

$d$

$\Gamma$

$\{\gamma\}_{\Gamma}$

$\gamma=\gamma_{1}\cdots\gamma_{d}$

$\cdots$

$\gamma_{d}\in\Gamma^{1}$

$7\iota.\gamma_{2},$

$\cdots,$

$\gamma_{1},$

$\gamma_{d-1}\cdot\gamma_{d}$

$l(\gamma\iota^{-1}\gamma\gamma_{1})=l(\gamma_{2}\cdots\gamma_{d}\gamma_{1})0$ , then we have $|S|=q^{k-1}(q+1)$ . In this case, for each $(mod d)$ , let are ﬀee and be a subset of formed of all $x\in S$ such that $q^{k-1}(q-1)$ (see Lemma remark, Then, elements consists of by the previous products. 5 . If $k=0$ , we simply put $S_{i}=S=\{I\}(1\leq i\leq d)$ . We shall prove that the following set of $dq^{k-1}(q-1)(k>0)$ or $d(k=0)$ elements of forms a set of representatives of all ; -conjugacy classes contained in $\pi_{b}\neq\pi_{f}$

$\gamma_{d}\cdot\gamma_{1}$

$ x\in\Gamma$

$=u\pi_{i_{1}}\cdots\pi_{i_{l}}$

$\pi_{i_{l}}\neq\pi_{b};\gamma_{d}\cdot x^{-1}$

$x\cdot\gamma_{1}$

$\pi_{i_{l}}\neq\pi_{f}$

$x\cdot\gamma_{1},$

$i$

$\gamma_{d}\cdot x^{-1}$

$\gamma_{i}\cdot\gamma_{i+1}$

$\gamma_{d},$

$\gamma_{1}$

$\gamma_{i},$

$\gamma_{i+1}$

$k$

$S$

$k$

$k$

$\Gamma^{0}\backslash \Gamma^{k/2}$

$i$

$S$

$S_{i}$

$\gamma_{i-1}\cdot x^{-1}$

$x\cdot\gamma_{i}$

$S_{i}$

$)$

$\Gamma$

$\Gamma^{0}$

$\{\gamma^{r}\}_{\Gamma}\cap\Gamma^{dr+k}$

(79)

$\left\{\begin{array}{ll}y_{1}(\gamma_{1}\gamma_{2}\cdots\gamma_{d})^{r}y_{1}^{-1}; & y_{1}\in S_{1}\\y_{2}(\gamma_{2}\gamma_{3}\cdots\gamma_{1})^{r}y_{2}^{-1}; & y_{2}\in S_{2}\\:

& \end{array}\right.$

: $y_{d}(\gamma_{d}\gamma_{1}\cdots\gamma_{d-1})^{r}y_{d}^{-1}$

;

$y_{d}\in S_{d}.$

Since the products are free, the product is free (corollary ofLemma 9); hence they are contained in

$y_{i}(\gamma_{i}\cdots\gamma_{i-1})^{r}y_{i}^{-1}=y_{i}\cdot\gamma_{i}\cdots\cdot\cdot\gamma_{i-1}\cdot y_{i}^{-1}$

$\gamma_{i-1}\cdot y_{i}^{-1}$

$y_{i}\cdot\gamma_{i},$

$\Gamma^{dr+k}$

. On the other hand, since

$(\gamma_{i}\gamma_{i+1}\cdots\gamma\vdash 1)^{r}=(\gamma_{1}\cdots\gamma_{i-1})^{-1}\gamma^{r}(\gamma_{1}\cdots\gamma\leftarrow 1)$

they are contained in First, let us prove that the distinct members of (79) are not other. Suppose that

,

$\{\gamma^{r}\}_{\Gamma}.$

$\Gamma^{0}$

-conjugate with each

$y_{i}(\gamma_{i}\cdots\gamma_{i-1})^{r}y_{i}^{-1}=uy_{j}’(\gamma_{j}\cdots\gamma_{j-1})^{r}y_{J^{-1}}’u^{-1}$

holds with

, and

$u\in\Gamma^{0},1\leq j\leq i\leq d$

$y_{i}\in S_{i},$

$y_{j}’\in S_{j}$

. Then, this implies that

is primitive (it is . Since commutes with conjugate to ), its centralizer in is the free cyclic group generated by itself. Hence, we get $(\gamma_{i}\cdots\gamma_{i-1})^{r}$

$y_{i}^{-1}uy_{j}’(\gamma_{J}\gamma_{j+1}\cdots\gamma_{i-1})$

$\Gamma-$

$\gamma_{i}\cdots\gamma_{i-1}$

$\Gamma$

$\gamma$

$y_{i}^{-1}uy_{j’}(\gamma_{\dot{j}}\gamma_{j+1}\cdots\gamma_{i-1})=(\gamma_{i}\cdots\gamma_{i-1})^{s}$

with some $s\in Z$ ; hence we get (80)

$uy_{j’}(\gamma_{j}\gamma_{j+1}\cdots\gamma_{i-1})=y_{i}(\gamma_{\iota}\gamma_{i+1}\cdots\gamma_{i-1})^{S} (s\in Z)$

.

appears, $ifs1$ , let be a prime element of that by ; if $e=1$ and $f>1$ , let be any element of outside and put and put . Then in either case the series has the property: if for some $u\in U_{p}$ , then $i=j$ . From this follows immediately $k_{\mathfrak{p}}\neq Q_{p},$

$V_{\mathfrak{p}}$

$V_{\mathfrak{p}}=\mathcal{B}_{\mathfrak{p}}\cap PSL_{2}(O_{\mathfrak{p}})$

$\mathcal{B}_{\mathfrak{p}}/\mathcal{B}_{p}$

$\mathcal{B}_{\mathfrak{p}},$

$\mathcal{B}_{\mathfrak{p}}/\mathcal{B}_{p}$

$|V_{\mathfrak{p}}\backslash \mathcal{B}_{\mathfrak{p}}/\mathcal{B}_{p}|=\infty$

$V_{\mathfrak{p}}\backslash \mathcal{B}_{\mathfrak{p}}/\mathcal{B}_{p}$

$k_{\mathfrak{p}}\neq Q_{p}$

$k_{\mathfrak{p}}\neq Q_{p}$

$e$

$k_{\mathfrak{p}}/Q_{p}$

$ef=[k_{\mathfrak{p}} :

Q_{p}]>1$

$\omega_{i}=\pi^{-(ie+1)}(i\geq 1)$

$\omega_{i}=\omega p^{-i}(i\geq 1)$

$\pi$

$\omega$

$k_{\mathfrak{p}}$

$O_{\mathfrak{p}}$

$\mathfrak{p}+Z_{p},$

$\omega_{i}(i\geq 1)$

$\omega_{f}-u\omega_{j}\in O_{\mathfrak{p}}+Q_{p}$

that the elements $|V_{\mathfrak{p}}\backslash \mathcal{B}_{\mathfrak{p}}/\mathcal{B}_{p}|=\infty.$

$(_{0}^{1}\omega_{1^{t\rangle}}(i\geq 1)$

belong to the diﬀerent double cosets

$V_{\mathfrak{p}}\backslash \mathcal{B}_{\mathfrak{p}}/\mathcal{B}_{p}$

; hence $\square .$

CHAPTER 1.

2. DETAILED STUDY OF ELEMENTS OF

35

$\Gamma$

\S 28. Proof of Theorem 3. The notations being as in \S 25, let be the parabolic R$ $G_{R,z}\cong ; hence stabilizer of in . Then and is abelian, and if is any element of , then with is the centralizer of in . We shall $G_{R,z}$

$H^{0}$

$H^{0}=\{\gamma\in\Gamma|\gamma_{R}\in G_{Rz}\}$

$G_{R}$

$z$

$H^{0}$

$\xi$

$H^{0}$

$\xi\neq 1$

$\xi$

$\Gamma$

prove (108)

$(H:H^{0})=\infty.$

For this purpose, let

be an open compact subgroup of and put Then by Lemma 14 applied to , we conclude that is a cusp of ; and hence for any is also a cusp of . But since there are only finitely many non-equivalent cusps of , we have $V$

$\Delta=\Gamma_{R}^{V}$

$\gamma\in\Gamma,$

$\Gamma^{V}=\Gamma\cap(G_{R}\times V\gamma.$

$G_{\mathfrak{p}}$

$z$

$\gamma=\epsilon_{R}$

$\Gamma_{R}^{V}$

$\Gamma_{R}^{V}$

$\gamma_{R}z$

$\Gamma_{R}^{V}$

(109)

$|\Gamma^{V}\backslash \Gamma/H|1,$ $N\not\equiv 0(mod p)$ be an integer, and let principal congruence subgroup of ;

,

$(\Gamma^{\prime 0})_{R}.$

$g’$

$\Gamma’=\Gamma(N\gamma$

be the

$\Gamma$

(209)

$\Gamma(N)=\{\gamma\in SL_{2}(Z^{(p)})|\gamma\equiv\pm 1(modN)\}/\pm 1 (N>1)$

.

Put $n=(\Gamma:\Gamma(N))=\left\{\begin{array}{ll}6 & (N=2) ,\\\frac{N^{3}}{2}\prod_{\eta_{N}}(1_{7^{1}}-) & (N>2) ;\end{array}\right.$

put $s=n/N$, and let be the smallest positive integer such that $ p^{d}\equiv\pm 1(mod N\gamma$ . Then we have $d$

(210)

The genus

$\prod_{P\in p_{\infty}(\Gamma)},(1-u^{\deg P})=(1-u^{d})^{s/d}.$

is given by $g’=\frac{N-6}{12N}n+1$ , and since $H’=(\rho-1)(g’-1+\frac{s}{2})=\frac{n}{12}(p-1)$ . Hence (211)

$g’$

of

$\Gamma(N)$

$\Gamma(N)$

is torsion-ﬀee, we have

$\zeta_{\Gamma’}(u)\times(1-u^{d})^{-s/d}=\frac{P’(u)}{(1-u)(1-p^{2}u)}\times(1-u)^{\tau^{n}z^{(p-1)}}.$

CHAPTER 2

Introduction to Part 1 and Part 2.

Chapter 2 consists of two parts, Part 1 (\S 1-\S 17) and Part 2 (\S 18-\S 36). The subject -field”, where here is what we call . The definition is as follows. $A$ field is called a -field over a subfield if $\dim_{k}L=1$ and if acts eﬀectively on as a group of field automorphisms over , fulfilling the following conditions (i) For each open compact subgroup , its fixed field is finitely generated over , and $L/L_{V}$ is nomlally and separably algebraic. Moreover, is topologically isomorphic to the Krull’s Galois group of $L/L_{V}.$ (ii) Almost all prime divisors of over are unramified in (iii) The fixed field of is ( is called the onstant field of ) The motivation for the study of such a field is this: If is a discrete subgroup of $G=G_{R}\times G_{p}$ with finite-volume-quotient such that the projections are dense in respectively, then defines a -field over the complex number field , and conversely (Theorem 1, \S 9). Thus and (over C) are equivalent notions. Moreover, it seems that the study of -fields over algebraic number fields 2 is crucial for the solution of our problems. Thus we meet our first problem: every -field over a constantfield extension -field over an algebraic numberfield?” This problem is solved aﬃrmatively in Part 2 (Theorem 4, \S 18). The readers note, however, that this would not be remarkable enough without ”essential uniquenes , which is guaranteed by Theorems 5, 6, 7 (\S 18, \S 32, \S 33) under a certain condition on . Namely, by Theorem 5, under a condition on which is always satisfied if is maximal (see \S 10), there is a unique over an algebraic number field such -field ”

$a$

$L$

$G_{\mathfrak{p}}=PSL_{2}(k_{\mathfrak{p}})$

$G_{\mathfrak{p}}$

$k$

$G_{\mathfrak{p}}$

$L$

$G_{p}$

$1_{:}$

$k$

$L_{V}$

$V\subset G_{\mathfrak{p}}$

$V$

$k$

$k$

$G_{\mathfrak{p}}$

$L.$

$k$

$L_{V}$

$k$

$L.$

$c$

$\Gamma$

–

$\Gamma_{R},$

$G_{R},$

$\Gamma_{\mathfrak{p}}$

$\Gamma$

$G_{\mathfrak{p}}$

$L$

$G_{\mathfrak{p}}$

$C$

$L$

$\Gamma$

$G_{\mathfrak{p}}$

’

$G_{\mathfrak{p}}$

$L$

$C$

$Is$

$L_{0}$

$ofaG_{\mathfrak{p}}$

$s$

$ofL_{0}$ $L$

$L$

$3G_{\mathfrak{p}}$

$\Gamma$

$k_{0}$

$L_{k_{\theta}}$

that (i) is a constant field extension of , and (ii) if $L$ is a constant field extension of another $L$

$L_{b}$

$G_{\mathfrak{p}}$

-field

contains and Thus if is maximal, then defines a unique -field . Theorems 6, 7 are some variations of Theorem 5. $k_{0}$

$\Gamma$

$L_{k}$

over a field

$k\subset C$

, then

$k$

$L_{k}=L_{h}\cdot k.$

$\Gamma$

$G_{\mathfrak{p}}$

$L_{k_{0}}$

over an algebraic number field

$k_{0}$

lSee also \S 1. We do not assume that

is the full automorphism group of over $2By$ an algebraic number field, we always mean a finite extension of the field ofrationals is unique not only up to isomorphisms, but also as a -invariant subfield of $L$

$G_{\mathfrak{p}}$

$3L_{\hslash_{0}}$

$G_{\mathfrak{p}}$

63

$k.$

$L.$

$Q.$

64

In the last two sections (\S 35, \S 36), we shall prove that under a certain condition on (to which no counterexample is known), the field $h$ contains the field $F$ defined by $F=Q((tr\gamma_{R})^{2}|h\in\Gamma_{R})$ (Theorem 8, \S 36). The idea of ﬃe proof is $\Gamma$

(i) to consider the -fixed points (on ) and the rotation arguments ofthe stabilizers algebraic terms, and (ii) to prove that $F$ is generated over $Q$ only by of elliptic elements of $\mathfrak{H}$

$\Gamma_{R}$

$(tr\gamma_{R})^{2}$

$\gamma_{R}$

$in$

$\Gamma_{R}.$

The proof of (ii) is given in Chapter 3 (\S 11). $A$ further study of the relations between , and will be left to the next stage ofthis chapter.

$k_{0},F$

$Q((tr\gamma_{\mathfrak{p}})^{2}|\gamma_{\mathfrak{p}}\in\Gamma_{\mathfrak{p}})$

Part 1 is rather a preliminary to Part 2. In Theorem 1 (\S 9), the one-to-one correspon(over C) is established. In Theorem 2 (\S 10), some Galois theoly” between dence and is proved.4 In particular, it is shown that is irreducible (see \S 10) if and only if is of finite index in $Aut_{C}L$ , a fact is maximal. In Theorem 3 (\S 11), it is proved that needed in Part 2. A large part of Part 2 is devoted to the proof of Theorem 4 $(i.e., \S 21\sim\S 31)$ . Two basic lemmas for this proof are : $\Gamma\leftrightarrow L$

$L$

$L$

$\Gamma$

$\Gamma$

$G_{\mathfrak{p}}$

(i) is a certain free product wrtb amalgamation (Lemma 7, \S 28), and (ii) homomorphisms of satisfying some conditions are induced by inner auinto tomorphisms $ofG_{R}’=PL_{2}(R)$ (Lemma 8, \S 29). $G_{\mathfrak{p}}$

$G_{R}$

$\Gamma_{R}$

As an example of

$G_{\mathfrak{p}}$

we shall treat the -field over that corresponds to (see \S 2). This field is treated in connection with Theorems 1, 3

-fields,

$G_{p}$

$L$

$C$

the group $\Gamma=PSL_{2}(Z^{(p)})$ and 5, in \S 2, \S 17 and \S 34, respectively. Although the -field” can be defined for any locally compact, non-compact, and totally disconnected group , our main results after \S 11 are essentially based on the (see Lemmas 1, 4 and 6). Moreover, the particular structure of the group with only examples of -fields that we know at present are those for ; and for such , we can obtain results similar to ours immediately from our results (e.g., by using Proposition 4 (\S 12) and Theorem 6 (\S 32)). Therefore, we is the group . shall assume throughout the chapter that $G_{\mathfrak{p}}$

$G_{\mathfrak{p}}$

$G_{\mathfrak{p}}=PSL_{2}(k_{\mathfrak{p}})$

$G_{\mathfrak{p}}\supset PSL_{2}(k_{\mathfrak{p}})$

$G_{\mathfrak{p}}$

$(G_{p} :

PSL_{2}(k_{\mathfrak{p}}))