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E-raamat: Introduction To Non-abelian Class Field Theory, An: Automorphic Forms Of Weight 1 And 2-dimensional Galois Representations

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(Waseda Univ, Japan), (Hosei Univ, Japan)
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Introduction To Non-abelian Class Field Theory, An: Automorphic Forms Of Weight 1 And 2-dimensional Galois RepresentationsSuurem pilt
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This monograph provides a brief exposition of automorphic forms of weight 1 and their applications to arithmetic, especially to Galois representations. One of the outstanding problems in arithmetic is a generalization of class field theory to non-abelian Galois extension of number fields. In this volume, we discuss some relations between this problem and cusp forms of weight 1.
Preface vii
Part I
1(90)
1 Higher reciprocity laws
3(18)
1.1 Some examples of non-abelian case
4(11)
1.1.1 f(x) = x3 -- d
4(5)
1.1.2 f(x) = 4x3 -- 4x2 + 1
9(4)
1.1.3 f(x) = x4 -- 2x2 + 2
13(2)
1.2 Modular forms and Hecke operators
15(6)
1.2.1 SL2(Z) and its congruence subgroups
15(1)
1.2.2 The upper half-plane
16(1)
1.2.3 Modular forms and cusp forms
17(1)
1.2.4 Hecke operators
18(3)
2 Hilbert class fields over imaginary quadratic fields
21(16)
2.1 The classical theory of complex multiplication
21(3)
2.2 Proof of Theorem 2.1
24(5)
2.3 Schlafli's modular equation
29(1)
2.4 The case of q = 47
30(7)
3 Indefinite modular forms
37(30)
3.1 Hecke's indefinite modular forms of weight 1
38(1)
3.2 Ray class fields over real quadratic fields
38(2)
3.3 Positive definite and indefinite modular forms of weight 1
40(4)
3.4 Numerical examples
44(7)
3.5 Higher reciprocity laws for some real quadratic fields
51(2)
3.6 Cusp forms of weight 1 related to quartic residuacity
53(4)
3.7 Fundamental lemmas
57(3)
3.8 Three expressions of θ(τ; K)
60(7)
4 Dimension formulas in the case of weight 1
67(24)
4.1 The Selberg eigenspace M(κ, λ)
67(4)
4.2 The compact case
71(5)
4.3 The Arf invariant and d1 mod 2
76(6)
4.3.1 The Arf invariant of quadratic forms mod 2
76(2)
4.3.2 The Atiyah invariant on spin structures
78(2)
4.3.3 The Arf invariant and d1 mod 2
80(2)
4.4 The finite case 1 (: Γ Э --I)
82(4)
4.5 The finite case 2 (: Γ Э --I)
86(2)
4.6 The case of Γ0(p)
88(3)
Part II
91(56)
5 2-dimensional Galois representations of odd type and non-dihedral cusp forms of weight 1
93(14)
5.1 Galois representations of odd type
93(5)
5.1.1 Artin L-functions and the Artin conjecture
93(1)
5.1.2 2-dimensional Galois representations of odd type and the Langlands program
94(4)
5.2 The case of types A4 and S4: Base change theory
98(3)
5.2.1 Results of Serre-Tate
98(1)
5.2.2 Base change for GL2
98(1)
5.2.3 The case of types A4 and S4
99(2)
5.3 The case of type A5
101(2)
5.3.1 The first example due to Buhler
101(1)
5.3.2 Icosahedral Artin representations
102(1)
5.4 The Serre conjecture
103(1)
5.5 The Stark conjecture in the case of weight 1
104(3)
5.5.1 The Stark conjecture
104(1)
5.5.2 The value of L (1/2, ε)
105(2)
6 Maass cusp forms of eigenvalue 1/4
107(8)
6.1 Maass cusp forms and Galois representations of even type
107(3)
6.1.1 Maass forms of weight zero
107(1)
6.1.2 Maass forms with weight
108(1)
6.1.3 Galois representations of even type
109(1)
6.2 Automorphic hyperfunctions of weight 1
110(5)
6.2.1 Limits of discrete series
110(1)
6.2.2 Automorphic hyperfunctions of weight 1
110(5)
7 Selberg's eigenvalue conjecture and the Ramanujan-Petersson conjecture
115(10)
7.1 Five conjectures in arithmetic
115(6)
7.1.1 Selberg's eigenvalue conjecture (C1)
115(1)
7.1.2 The Sato-Tate conjecture (C2)
116(4)
7.1.3 The Ramanujan-Petersson conjecture (C3)
120(1)
7.1.4 Linnik-Selberg's conjecture (C4)
121(1)
7.1.5 The Gauss-Hasse conjecture (C5)
121(1)
7.2 Some relations between the five conjectures
121(4)
7.2.1 Conjectures C1 and C3
121(1)
7.2.2 Conjectures C1 and C5
122(1)
7.2.3 Conjectures C3 and C4
123(1)
7.2.4 Conjectures C2 and C3
124(1)
8 Indefinite theta series
125(6)
8.1 Indefinite quadratic forms and indefinite theta series
125(6)
8.1.1 Hecke's indefinite theta series
125(1)
8.1.2 Polishchuk's indefinite theta series
126(5)
9 Hilbert modular forms of weight 1
131(16)
9.1 Hilbert modular forms
131(3)
9.1.1 Hilbert modular groups
131(1)
9.1.2 Hilbert modular forms
132(2)
9.2 A dimension formula for the space of the Hilbert cusp forms of weight 1 of two variables
134(13)
9.2.1 Introduction
134(2)
9.2.2 Fundamental lemma
136(3)
9.2.3 Modified trace formula
139(3)
9.2.4 Eisenstein series attached to ∞
142(2)
9.2.5 The trace at the cusp
144(3)
Appendix. Some dimension formula and traces of Hecke operators for cusp forms of weight 1 -- Gottingen talk,
1989. By Toyokazu Hiramatsu
147(18)
§ 1 Introduction
147(1)
§ 2 Results
148(3)
§ 3 The Selberg eigenspace
151(1)
§ 4 The compact case
152(4)
§ 5 The finite case 1: Γ Э --I
156(3)
§ 6 The finite case 2: Γ Э --I
159(2)
§ 7 The case of Γ0(p)
161(2)
§ 8 Trace of Hecke operators in the case of weight 1
163(2)
Bibliography 165(8)
Index 173
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