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Diamond F., Shurman J. — First Course in Modular Forms
Diamond F., Shurman J. — First Course in Modular Forms

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Название: First Course in Modular Forms

Авторы: Diamond F., Shurman J.


This book introduces the theory of modular forms with an eye toward the Modularity Theorem:

All rational elliptic curves arise from modular forms.

The topics covered include:

- elliptic curves as complex tori and as algebraic curves

- modular curves as Riemann surfaces and as algebraic curves

- Hecke operators and Atkin-Lehner theory

- Hecke eigenforms and their arithmetic properties

- the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms

- elliptic and modular curves modulo p and the Eichler-Shimura Relation

- the Galois representations associated to elliptic curves and to Hecke eigenforms

As it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their applications to number theory.

A First Course in Modular Forms is written for beginning graduate students and advanced undergraduates. It does not require background in algebraic number theory or algebraic geometry, and it contains exercises throughout.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2007

Количество страниц: 436

Добавлена в каталог: 19.06.2008

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Предметный указатель
$l$-adic cyclotomic character      379
$l$-adic integer      373
$l$-adic Tate module of a modular curve      386
$l$-adic Tate module of an elliptic curve      382
$l$-adic Tate module of the curve xy = 1      373
$l$-adic Tate module, associated to $A_f$      389
$\lambda$-adic integers      375
$\lambda$-adic numbers      375
Abelian variety associated to an eigenform      241
Abel’s theorem      84 214
Absolute Galois group of Q      376
Addition law of elliptic curves, algebraic      254 312
Addition law of elliptic curves, geometric      253 312
Adjoint      183
Admissible change of variable      251 311
Affine plane      252
Algebraic closure of a field      250
Algebraic curve, affine      258
Algebraic curve, nonsingular      258
Algebraic curves, isomorphic      263
Algebraic curves, isomorphic over k      264
Algebraic element over a field      250
Algebraic field extension      250
Algebraic integer      231
Algebraic number      230
Algebraically closed      231
Algebraically closed field      250
Artin’s conjecture      395
Automorphic form of weight k      72
Base point      214
Bernoulli number      9 134
Bernoulli number of $\psi$      135
Bernoulli polynomial      134
Bezout’s Theorem      252
Birational equivalence of algebraic curves      264
Birch and Swinnerton — Dyer conjecture      362
Canonical divisor on a Riemann surface      84
Commensurable congruence subgroups      164
Compatibility condition for meromorphic differentials on a Riemann surface      78
Complex multiplication      30
Complex torus      25
Conductor of a Dirichlet character      117
Conductor of a normalized eigenform      198
Conductor of an elliptic curve, algebraic      324
Conductor of an elliptic curve, analytic      292
Congruence subgroup      13
Conjugate function fields      264
Coordinate ring of an algebraic curve      257
Cubic Reciprocity Theorem      160
Curves–Fields Correspondence, Part 1      264
Curves–Fields Correspondence, Part 2      265
Cusp form of weight k      6
Cusp form of weight k with respect to a congruence subgroup      17
Cusp of a modular curve      58
Cusp of a modular curve, irregular      89
Cusp of a modular curve, regular      89
Cusp of an elliptic curve      314
Cyclic quotient isogeny      27
Decomposition group      369
Decomposition group, absolute      377
Decomposition index      367
Dedekind eta function      19
Degree of a map of Riemann surfaces      65
Degree of a morphism of algebraic curves      269
Degree of a morphism, inseparable      320
Degree of a morphism, separable      320
Degree-0 divisors of an algebraic curve      270
Desingularizing an algebraic curve      264
Dirichlet character, modulo N      116
Dirichlet character, primitive      117
Dirichlet character, trivial      116
Dirichlet L-function      121
Dirichlet’s Theorem on Arithmetic Progressions      370
Discriminant of a quadratic number field      367
Discriminant of a Weierstrass equation      250 310
Discriminant, function on $\mathcal{H}$      6
Divisor group of an algebraic curve      269
Divisor on a Riemann surface      83
Double coset      103 164
Double coset operator as map of Picard groups      227
Double coset operator on Jacobians      228
Dual isogeny      28
Eichler — Shimura relation      354
Eigenform      186 195
Eigenform, normalized      195
Eigenspace of $\mathcal{M}_k (\Gamma_1 (N ))$      119
Eisenstein series of weight 2      18
Eisenstein series of weight 2, normalized      19
Eisenstein series of weight k      4
Eisenstein series of weight k for a lattice      32
Eisenstein series of weight k, normalized      10
Elliptic curve over a field of characteristic 0      251
Elliptic curve over an arbitrary field      310
Elliptic curve, complex      35
Elliptic curve, ordinary      313
Elliptic curve, supersingular      313
Elliptic curve, universal      282 311
Elliptic point of a congruence subgroup      48
Enhanced elliptic curve for $\Gamma(N )$      38
Enhanced elliptic curve for $\Gamma_0 (N )$      37
Enhanced elliptic curve for $\Gamma_1 (N )$      37
Enhanced elliptic curve for $\Gamma_1 (N )$, algebraic      301
Enhanced elliptic curve for $\Gamma_1 (N )$, complex algebraic      301
Equivalence of Galois representations      379 380
Euler product      200
Exceptional points of a map      65
Factor of automorphy      14
Faltings’s Isogeny Theorem      362
Fermat’s Last Theorem      396
Field extension, inseparable      319
Field extension, purely inseparable      319
Field extension, separable      319
Fontaine — Mazur — Langlands Conjecture      395
Forward change of variable formula      218
Forward map of Jacobians      218
Forward map of Picard groups      220
Four squares problem      11
Fourier transform      9 143
Fourier transform, finite      150
Free Q-module on the points of a modular curve      86
Frobenius element of a Galois group      369
Frobenius element, absolute      377
Frobenius map on $\bar{F}_p$      317
Frobenius map on an algebraic curve      318
Function field of an algebraic curve      257
Function field over k      264
Functional equation of $\Gamma$      120
Functional equation of $\ksi(s)$      121
Functional equation of L(s, f )      204
Fundamental domain for $SL_2 (Z)$      54
Galois representation, $l$-adic      378 380
Galois representation, associated to $A_f$      389 390
Galois representation, associated to $X_1 (N )$      387 388
Galois representation, associated to an elliptic curve      383
Galois representation, associated to f      390 391
Galois representation, geometric      395
Galois representation, modular      391 395
Galois representation, odd      393
Galois representation, unramified      380
Gamma function      120
Gauss sum of a Dirichlet character      118
Generating function      11 134 179
Genus      65
Genus formula      68
Genus of a modular curve      62
Global minimal discriminant of an elliptic curve      323
Hecke algebra over C      234
Hecke algebra over Z      234
Hecke operator $T_p$      169
Hecke operator $T_p$, four compatible notions of      175
Hecke operator d (diamond operator)      169
Hecke operators on Jacobians      229
Hilbert basis theorem      266
Holomorphic at $\infty$      3 16
Homology group of a Riemann surface      213
Hurwitz zeta function      135
Hyperbolic measure on H      181
Igusa’s Theorem      348
Inertia group      369
Inertia group, absolute      377
Integrally closed      232
Invariant of a Weierstrass equation      250 310
Inverse limit      373
Inverse Mellin transform      147
Isogeny      27 244
Isomorphism of algebraic curves      263
Isotropy subgroup      48
Jacobian of a modular curve      227
Jacobian of a Riemann surface      214
K-points of E      254
Kronecker symbol      367
Krull intersection theorem      342
Krull topology      377
L-function of a modular form      200
L-function of an elliptic curve (Hasse — Weil)      361
Lattice in C      25
Legendre relation      138
Level of a congruence subgroup      13
Linear space of a divisor on a Riemann surface      84
Lipschitz formula      147
Local ring of an algebraic curve at a point      259
Local ring of the Hecke ring      239
Localization of $\bar{Z}$ at $\mathfrak{p}$      329
Localization of Z at p      334
Maximal ideal of the local ring at a point      259
Mellin transform      136 145 204
Meromorphic differential on a Riemann surface      78
mod $\mathscr{L}$ representation      385
mod $\mathscr{L}$ representation, modular      395
Modular curve      38
Modular form of weight k      4
Modular form of weight k with respect to a congruence subgroup      17
Modular function      7
Modular group      1
Modular invariant      7
Modular parameterization of an elliptic curve      63
Modularity Theorem, strong Version $A_Q$      362
Modularity Theorem, strong Version R      392
Modularity Theorem, Version $A_C$      246
Modularity Theorem, Version $a_p$      356
Modularity Theorem, Version $A_Q$      294
Modularity Theorem, Version $J_C$      215
Modularity Theorem, Version $J_Q$      293
Modularity Theorem, Version $X_C$      63
Modularity Theorem, Version $X_Q$      292
Modularity Theorem, Version L      362
Modularity Theorem, Version R      392
Moduli space      38
Moduli space for $\Gamma_1 (N )$, algebraic      301
Moduli space for $\Gamma_1 (N )$, complex algebraic      301
Moebius function      113
Mordell — Weil theorem      254
Morphism of algebraic curves      263
Morphism of algebraic curves, defined over a field      264
Morphism, inseparable      319
Morphism, purely inseparable      319
Morphism, separable      319
Multiplicity One property of newforms      196
Multiply-by-integer isogeny      27
N -torsion subgroup of a complex torus      30
N -torsion subgroup of an elliptic curve      254
n-dimensional projective space over K      255
Nakayama’s Lemma      235
NEW      173
Newform      195
Newforms at level N      188
Node of an elliptic curve      314
Noetherian ring      266
Nonsingular, geometrically      251
Norm map of function fields      219
Normal linear operator      183
Nth division polynomials of an elliptic curve      254
Number field      231
Number field of a modular form      234
Number ring of a field      232
OLD      173
Oldforms at level N      187
Orthogonality relations      117
Orthogonality relations, modified versions      118
p-adic valuation      322
Perfect field      321
Perfect pairing      243
Period of an elliptic point      49
Petersson inner product      182
Picard group of a Riemann surface      214
Picard group of an algebraic curve      270
Poisson summation formula      9 144
Polynomial function on an algebraic curve      257
Power sum      133
Prime subfield of a field      312
Principal congruence subgroup of level N      13
Principal divisor      270
Profinite      374
Projective line      256
Projective plane over k      252
Pullback of function fields      217
Pullback of function fields of algebraic curves      265
Pullback of holomorphic differentials      218
Pullback of meromorphic differentials      77
Ramification degree      367
Ramification degree of a map of Riemann surfaces      65
Ramification degree of a morphism at a point      269
Ramified prime in a number field      367
Rational function defined over a field      263
Rational function on an algebraic curve      257
Rational integer      231
Rational map of elliptic curves      264
Reduction of a projective algebraic curve over Q at p      338
Reduction of an affine algebraic curve over Q at p      337
Reduction of an elliptic curve over Q, bad      323
Reduction of an elliptic curve over Q, bad, additive      323
Reduction of an elliptic curve over Q, bad, multiplicative      323
Reduction of an elliptic curve over Q, good      323
Reduction of an elliptic curve over Q, good, ordinary      323
Reduction of an elliptic curve over Q, good, supersingular      323
Representation number      11
Residue degree      367
Residue field      367
Reverse change of variable formula      221
Reverse map of Jacobians      222
Reverse map of Picard groups      224
Riemann surface      25 47
Riemann zeta function      4 121
Riemann — Hurwitz formula      66
Riemann — Roch theorem      84
Schwartz function      151
Serre’s conjecture      396
Strong Multiplicity One property of newforms      198
Sylvester matrix      232
Tangent line to an algebraic curve      259
Tchebotarov Density Theorem      371
Tensor product of a finitely generated Abelian group and a field      240
Theta function      143
Trace map of holomorphic differentials      221
Transcendental element over a field      250
Uniformizer at a point      260
Upper half plane      2
Valuation at a point      261
Variety      293
Weakly modular function of weight k      2
1 2
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