Preface Modular Forms, Elliptic Curves, and Modular Curves ... 1.1 First definitions and examples 1.2 Congruence subgroups 1.3 Complex tori 1.4 Complex tori as elliptic curves 1.5 Modular curves and moduli spaces 2 Modular Curves as Riemann Surfaces 2.1 Topology 2.2 Charts 2.3 Elliptic points 2.4 Cusps 2.5 Modular curves and Modularity 3 Dimension Formulas 3.1 The genus 3.2 Automorphic forms 3.3 Meromorphic differentials 3.4 Divisors and the Riemann-Roch Theorem 3.5 Dimension formulas for even k 3.6 Dimension formulas for odd k 3.7 More on elliptic points 3.8 More on cusps 3.9 More dimension formulas 4 Eisenstein Series 4.1 Eisenstein series for SL2(Z) 4.2 Eisenstein series for F(N) when k≥3 4.3 Dirichlet characters, Gauss sums, and eigenspaces 4.4 Gamma, zeta, and L-functions 4.5 Eisenstein series for the eigenspaces when k≥3 4.6 Eisenstein series of weight 2 4.7 Bernoulli numbers and the Hurwitz zeta function 4.8 Eisenstein series of weight 1 4.9 The Fourier transform and the Mellin transform 4.10 Nonholomorphic Eisenstein series 4.11 Modular forms via theta functions 5 Hecke Operators 5.1 The double coset operator 5.2 The and Tp operators 5.3 The (n> and Tn operators 5.4 The Petersson inner product 5.5 Adjoints of the Hecke Operators 5.6 Oldforms and Newforms 5.7 The Main Lemma 5.8 Eigenforms 5.9 The connection with L-functions 5.10 Functional equations. 5.11 Eisenstein series again 6 Jacobians and Abelian Varieties 6.1 Integration, homology, the Jacobian, and Modularity 6.2 Maps between Jacobians 6.3 Modular Jacobians and Hecke operators 6.4 Algebraic numbers and algebraic integers 6.5 Algebraic eigenvalues 6.6 Eigenforms, Abelian varieties, and Modularity 7 Modular Curves as Algebraic Curves 7.1 Elliptic curves as algebraic curves 7.2 Algebraic curves and their function fields 7.3 Divisors on curves 7.4 The Weil pairing algebraically 7.5 Function fields over C 7.6 Function fields over Q 7.7 Modular curves as algebraic curves and Modularity 7.8 Isogenies algebraically 7.9 Hecke operators algebraically 8 The Eichler-Shimura Relation and L-functions 8.1 Elliptic curves in arbitrary characteristic 8.2 Algebraic curves in arbitrary characteristic 8.3 Elliptic curves over Q and their reductions …… 9 Galois Representations Hints and Answers to the Exercises List of Symbols Index References
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