Table of Contents
Preface vii
Part I 1
1 Higher reciprocity laws 3
1.1 Some examples of non-abelian case 4
1.1.1 f(x) = x3 - d 4
1.1.2 f(x) = 4x3 - 4x2 + 1 9
1.1.3 f(x) = x4 - 2x2 + 2 13
1.2 Modular forms and Hecke operators 15
1.2.1 SL2 (Z) and its congruence subgroups 15
1.2.2 The upper half-plane 16
1.2.3 Modular forms and cusp forms 17
1.2.4 Heckc operators 18
2 Hilbert class fields over imaginary quadratic fields 21
2.1 The classical theory of complex multiplication 21
2.2 Proof of Theorem 2.1 24
2.3 Schläfli's modular equation 29
2.4 The ease of q = 47 30
3 Indefinite modular forms 37
3.1 Hecke's indefinite modular forms of weight 1 38
3.2 Ray class fields over real quadratic fields 38
3.3 Positive definite and indefinite modular forms of weight 1 40
3.4 Numerical examples 44
3.5 Higher reciprocity laws for some real quadratic fields 51
3.6 Cusp forms of weight 1 related to quartic residuacity 53
3.7 Fundamental lemmas 57
3.8 Three expressions of θ(τ; K) 60
4 Dimension formulas in the case of weight 1 67
4.1 The Selberg eigenspace m(k, λ) 67
4.2 The compact case 71
4.3 The Arf invariant and d1 mod 2 76
4.3.1 The Arf invariant of quadratic forms mod 2 76
4.3.2 The Atiyah invariant on spin structures 78
4.3.3 The Arf invariant and d1 mod 2 80
4.4 The finite case 1 (: Γ ∋ -I) 82
4.5 The finite ease 2 (: Γ ∋ -I) 86
4.6 The case of Γ0(p) 88
Part II 91
5 2-dimensional Galois representations of odd type and non-dihedral cusp forms of weight 1 93
5.1 Galois representations of odd type 93
5.1.1 Artin L-functions and the Artin conjecture 93
5.1.2 2-dimensional Galois representations of odd type and the Langlands program 94
5.2 The case of types A4 and S4: Base change theory 98
5.2.1 Results of Serre-Tate 98
5.2.2 Base change for GL2 98
5.2.3 The case of types A4 and S4 99
5.3 The case of type A5 101
5.3.1 The first example due to Buhler 101
5.3.2 Icosahedral Artin representations 102
5.4 The Serre conjecture 103
5.5 The Stark conjecture in the case of weight 1 104
5.5.1 The Stark conjecture 104
5.5.2 The value of L (1/2, ε) 105
6 Maass cusp forms of eigenvalue 1/4 107
6.1 Maass cusp forms and Galois representations of even type 107
6.1.1 Maass forms of weight zero 107
6.1.2 Maass forms with weight 108
6.1.3 Galois representations of even type 109
6.2 Automorphic hyperfunctions of weight 1 110
6.2.1 Limits of discrete series 110
6.2.2 Automorphic hyperfunctions of weight 1 110
7 Selberg's eigenvalue conjecture and the Ramanujan-Petersson conjecture 115
7.1 Five conjectures in arithmetic 115
7.1.1 Selberg's eigenvalue conjecture (C1) 115
7.1.2 The Sato-Tate conjecture (C2) 116
7.1.3 The Ramanujan-Petersson conjectiue (C3) 120
7.1.4 Linnik-Selberg's conjecture (C4) 121
7.1.5 The Gauss-Hasse conjecture (C5) 121
7.2 Some relations between the five conjectures 121
7.2.1 Conjectures C1 and C3 121
7.2.2 Conjectures C1 and C5 122
7.2.3 Conjectures C3 and C4 123
7.2.4 Conjectures C2 and C3 124
8 Indefinite theta series 125
8.1 Indefinite quadratic forms and hide finite theta series 125
8.1.1 Hecke's indefinite theta series 125
8.1.2 Polishchuk's indefinite theta series 126
9 Hilbert modular forms of weight 1 131
9.1 Hilbert modular forms 131
9.1.1 Hilbert modular groups 131
9.1.2 Hilbert modular forms 132
9.2 A dimension formula for the space of the Hilbert cusp forms of weight 1 of two variables 134
9.2.1 Introduction 134
9.2.2 Fundamental lemma 136
9.2.3 Modified trace formula 139
9.2.4 Eisenstein series attached to ∞ 142
9.2.5 The trace at the cusp 144
Appendix: Some dimension formula and traces of Hecke operators for cusp forms of weight 1 - Göttingen talk, 1989. Toyokazu Hiramatsu 147
§1 Introduction 147
§2 Results 148
§3 The Selberg eigenspace 151
§4 The compact case 152
§5 The finite case 1: Γ ∋ -I 156
§6 The finite case 2: Γ ∋ -I 159
§7 The case of Γ0(p) 161
§8 Trace of Hecke operators in the case of weight 1 163
Bibliography 165
Index 173
