The Taniyama-Shimura-Weil conjecture

312 Pages·2009·7.735 MB·English

by  Adrien Lücker, Alex Monnard, Vlad Serban et al.

The Taniyama-Shimura-Weil conjecture
Adrien L(cid:252)cker - Alex Monnard
Vlad Serban - Daniele Rotanzi
Fabrizio Gelsomino - Kevin Fournier
Semester project of mathematics
Under the direction of Prof. Philippe Michel, Prof. Donna Testerman and
Prof. Eva Bayer

J’ai une dØmonstration vØritablement merveilleuse de cette proposition, que
cette marge est trop Øtroite pour contenir.
Pierre de Fermat (XVIIe siŁcle, France).
On entre dans la premiŁre chambre et elle est obscure. ComplŁtement
obscure. On se heurte aux meubles, on (cid:28)nit par conna(cid:238)tre leur
emplacement. AprŁs quelques six mois, on (cid:28)nit par trouver le commutateur
et soudain, la piŁce est ØclairØe. On peut voir exactement oø l’on se trouve.
Puis on passe (cid:224) la piŁce suivante, et l’on a(cid:27)ronte de nouveau six mois
d’obscuritØ. Donc, chacune des percØes qui ont ØtØ faites et qui sont parfois
brŁves, ne durant qu’un jour ou deux, sont l’accomplissement des mois de
t(cid:226)tonnements dans le noir, sans lesquels il n’y aurait jamais eu de lumiŁre.
Andrew Wiles (1953, USA).
i

Contents
List of Notation vii
I Complex Multiplication and Galois Theory 1
Introduction 3
1 Basic Galois theory 5
1.1 Basic de(cid:28)nitions . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Galois theory on number (cid:28)elds . . . . . . . . . . . . . . . . . . 12
Q
1.3 Finite abelian Galois extensions of . . . . . . . . . . . . . . 24
2 Elliptic curves 29
2.1 Genaral de(cid:28)nitions for elliptic curves . . . . . . . . . . . . . . 29
2.2 Elliptic curves in characteristic (cid:54)= 2,3 . . . . . . . . . . . . . . 34
2.3 Elliptic curves as a group . . . . . . . . . . . . . . . . . . . . . 39
2.4 Equivalence and isomorphism . . . . . . . . . . . . . . . . . . 44
C
2.5 Elliptic curves over . . . . . . . . . . . . . . . . . . . . . . . 58
3 Elliptic curves and Galois theory 75
3.1 Points of (cid:28)nite order . . . . . . . . . . . . . . . . . . . . . . . 75
3.2 Complex multiplication . . . . . . . . . . . . . . . . . . . . . . 86
3.3 The ring End(E) . . . . . . . . . . . . . . . . . . . . . . . . . 92
Conclusion 97
iii
CONTENTS
II Reduction Type and L-series of Elliptic Curves 99
Introduction 101
4 Good and Bad reduction of an elliptic curve 103
4.1 General Weierstrass equation . . . . . . . . . . . . . . . . . . 103
Q
4.2 Minimal Weierstrass equation and reduction over . . . . . 106
p
Q
4.3 Reduction type of elliptic curves over . . . . . . . . . . . . . 110
5 The L-series 115
5.1 Reminder on (cid:28)nite (cid:28)elds . . . . . . . . . . . . . . . . . . . . . 115
5.2 Several theorems on elliptic curves over (cid:28)nite (cid:28)elds . . . . . . 117
5.3 The zeta-function of an elliptic curve over (cid:28)nite (cid:28)elds . . . . . 119
Q
5.4 The L-series of an elliptic curve over . . . . . . . . . . . . . 124
6 Tate’s algorithm 127
6.1 Explanation of the Tate’s algorithm . . . . . . . . . . . . . . . 127
6.2 An application of Tate’s algorithm . . . . . . . . . . . . . . . 132
7 L-series and modular forms 135
III The Weil Conjectures for Elliptic Curves 139
Introduction 141
8 Algebraic Preliminaries 143
8.1 Galois Extensions . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.2 Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.3 Inseparable extensions . . . . . . . . . . . . . . . . . . . . . . 145
9 Algebraic Varieties and Curves 147
9.1 A(cid:30)ne and Projective Varieties . . . . . . . . . . . . . . . . . . 147
9.2 Algebraic Curves and their Maps . . . . . . . . . . . . . . . . 149
10 Elliptic Curves 151
10.1 The Dual Isogeny . . . . . . . . . . . . . . . . . . . . . . . . . 152
10.2 The Tate Module and Weil Pairing . . . . . . . . . . . . . . . 153
iv
CONTENTS
11 The Frobenius Map 155
11.1 Frobenius morphism for elliptic curves . . . . . . . . . . . . . 157
12 Hasse’s estimate and the Weil Conjectures 159
12.1 Hasse’s estimate . . . . . . . . . . . . . . . . . . . . . . . . . . 159
12.2 The Weil Conjectures . . . . . . . . . . . . . . . . . . . . . . . 161
IV Mordell-Weil Theorem 165
Introduction 169
13 Pro(cid:28)nite Groups 171
13.1 Projective and Direct limits . . . . . . . . . . . . . . . . . . . 171
13.2 Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
14 Cohomology and Discrete Valuation Rings 177
14.1 Group cohomology . . . . . . . . . . . . . . . . . . . . . . . . 177
14.2 Discrete Valuation Rings . . . . . . . . . . . . . . . . . . . . . 182
15 The Mordell-Weil Theorem 187
15.1 From Mordell-Weil to the Selmer group . . . . . . . . . . . . . 187
15.2 Finiteness of the Selmer group . . . . . . . . . . . . . . . . . . 189
15.3 The Descent Procedure . . . . . . . . . . . . . . . . . . . . . . 192
15.4 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . 194
V Modular Curves and Modularity Theorem 197
Introduction 199
16 The modular curve X(1) as a compact Riemann surface. 201
16.1 The modular group . . . . . . . . . . . . . . . . . . . . . . . . 201
16.2 The modular curve X(1) . . . . . . . . . . . . . . . . . . . . . 206
17 Modular curves as algebraic curves 215
17.1 Algebraic curves and function (cid:28)elds . . . . . . . . . . . . . . . 215
C
17.2 Function (cid:28)elds over . . . . . . . . . . . . . . . . . . . . . . 219
17.3 Modular curves as algebraic curves and modularity. . . . . . . 226
v
CONTENTS
VI Modular forms 229
Acknowledgements 231
Introduction 233
18 Prerequisites 235
19 Modular forms 239
19.1 Congruence subgroups . . . . . . . . . . . . . . . . . . . . . . 239
19.2 Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . 243
19.3 The Petersson inner product . . . . . . . . . . . . . . . . . . . 252
19.4 Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . 255
20 Hecke operators 261
20.1 The double coset operator and Hecke algebras . . . . . . . . . 261
20.2 Hecke operators for SL (Z) . . . . . . . . . . . . . . . . . . . . 267
2
20.3 Hecke operators for Γ (N) . . . . . . . . . . . . . . . . . . . . 270
0
20.4 Normality of Hecke operators . . . . . . . . . . . . . . . . . . 278
21 The L-function of a modular form 283
21.1 Euler product expansion of L(f,s) . . . . . . . . . . . . . . . 284
21.2 Oldforms and newforms . . . . . . . . . . . . . . . . . . . . . 287
21.3 Functional equation . . . . . . . . . . . . . . . . . . . . . . . . 295
Bibliography 298
Index 301
vi
List of Notation
We shall use these notations throughout this report:
E(k) k-rational points of an elliptic curve E.
k The algebraic closure of the (cid:28)eld k.
L/K Field extension K ⊂ L.
Gal(L/K) The Galois group of L/K.
G The n-torsion subgroup of the group G.
n
∆ The discriminant of an elliptic curve.
lim The direct limit.
−→
lim The projective limit.
←−
E(cid:101) The reduction of an elliptic curve E modulo p.
p
P2(k) The projective space over k.
∆(cid:101) The discriminant of a reduced elliptic curve modulo p.
p
v The p-adic valuation.
p
Z (T) The Z-function of E.
E
ζ (s) The zeta -function of E.
E
L (s) The L-function of E.
E
f(E) The exponent of the conductor.
v(D ) The valuation of the minimal discriminant of E.
E
P1(C) the Riemann sphere.
H the PoincarØ half-plane.
H∗ the extended upper half-plane.
L
the set of all latices.
Γ(1) the modular group.
p a prime number.
f| α(z) (detα)k/2j(α,z)−kf(αz).
k
ι the multiply-by-d map (cid:16)f(z) (cid:55)→ f(dz)(cid:17).
d
vii
LIST OF NOTATION
j(α,z) j(α,z) = cz +d for α = (a b) ∈ GL+(Q).
c d 2
M (Γ) the space of modular forms of weight k with respect to Γ.
k
S (Γ) the space of cusp forms of weight k with respect to Γ.
k
R(N) the Hecke algebra R(Γ (N),∆ (N)).
0 0
T , NT Hecke operator of level N.
n n
w the operator | ( −1).
N k N
viii