模形式基礎(chǔ)教程

內(nèi)容概要

　　《模形式基礎(chǔ)教程》是《數(shù)學(xué)研究生叢書》第228卷，內(nèi)容主要包括：橢圓曲線、復(fù)環(huán)面和代數(shù)曲，模曲線、黎曼曲面和代數(shù)曲線，Hecke算子和Athkin—Lehner理論，Hecke特征形式及它們的算術(shù)性質(zhì)，模曲線的雅可比行列式和Hecke特征形式的阿貝爾簇，橢圓曲線、模曲線模P及Eichler—Shimura關(guān)系，橢圓曲線和Hecke特征形式的Galois表示。

書籍目錄

PrefaceModular Forms， Elliptic Curves， and Modular Curves ...1.1 First definitions and examples1.2 Congruence subgroups1.3 Complex tori1.4 Complex tori as elliptic curves1.5 Modular curves and moduli spaces2 Modular Curves as Riemann Surfaces2.1 Topology2.2 Charts2.3 Elliptic points2.4 Cusps2.5 Modular curves and Modularity3 Dimension Formulas3.1 The genus3.2 Automorphic forms3.3 Meromorphic differentials3.4 Divisors and the Riemann-Roch Theorem3.5 Dimension formulas for even k3.6 Dimension formulas for odd k3.7 More on elliptic points3.8 More on cusps3.9 More dimension formulas4 Eisenstein Series4.1 Eisenstein series for SL2（Z）4.2 Eisenstein series for F（N） when k≥34.3 Dirichlet characters， Gauss sums， and eigenspaces4.4 Gamma， zeta， and L-functions4.5 Eisenstein series for the eigenspaces when k≥34.6 Eisenstein series of weight 24.7 Bernoulli numbers and the Hurwitz zeta function4.8 Eisenstein series of weight 14.9 The Fourier transform and the Mellin transform4.10　Nonholomorphic Eisenstein series4.11　Modular forms via theta functions5　Hecke Operators5.1 The double coset operator5.2 The and Tp operators5.3 The （n> and Tn operators5.4 The Petersson inner product5.5 Adjoints of the Hecke Operators5.6 Oldforms and Newforms5.7 The Main Lemma5.8 Eigenforms5.9 The connection with L-functions5.10 Functional equations.5.11 Eisenstein series again6　Jacobians and Abelian Varieties6.1 Integration， homology， the Jacobian， and Modularity6.2 Maps between Jacobians6.3 Modular Jacobians and Hecke operators6.4 Algebraic numbers and algebraic integers6.5 Algebraic eigenvalues6.6 Eigenforms， Abelian varieties， and Modularity7 Modular Curves as Algebraic Curves7.1 Elliptic curves as algebraic curves7.2 Algebraic curves and their function fields7.3 Divisors on curves7.4 The Weil pairing algebraically7.5 Function fields over C7.6 Function fields over Q7.7 Modular curves as algebraic curves and Modularity7.8 Isogenies algebraically7.9 Hecke operators algebraically8 The Eichler-Shimura Relation and L-functions8.1 Elliptic curves in arbitrary characteristic8.2 Algebraic curves in arbitrary characteristic8.3 Elliptic curves over Q and their reductions……9 Galois RepresentationsHints and Answers to the ExercisesList of SymbolsIndexReferences

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