Elliptic functions parametrize elliptic curves, and the intermingling of the analytic and algebraic-arithmetic theory has been at the center of mathematics since the early part of the nineteenth century. Some new techniques and outlooks have recently appeared on these old subjects, continuing in the tradition of Kronecker, Weber, Fricke, Hasse,Deuring. Shimura's book Introduction to the arithmetic theory of automorphic functions is a splendid modern reference, which I found very helpful myself to learn some aspects of elliptic curves. It emphasizes the direction of the Hasse-Weil zeta function, Hecke operators, and the generalizations due to him to the higher dimensional case (abelian varieties, curves of higher genus coming from an arithmetic group operating on the upper half plane, bounded symmetric domains with a discrete arithmetic group whose quotient is algebraic). I refer the interested reader to his book and the bibliography therein.
PART ONEccGENERAL THEORY Chapter1 Ellipti Functions 1 ThecLiouville Theorems 2 The Weierstrass Function 3 The AdditioncTheorem 4 Isomorphism Classescof Elliptic Curves 5 Endomorphisms and Automorphisms Chapter2 Homomorphisms 1 Points of Finite Order 2 Isogenies 3 The Involution Chapter 3 hecModular Function 1 The Modular Group 2 Automorphic Functions of Degree 2k 3 The Modular Functionj Chapter 4 Fourier Expansions 1 Expansion for Gk,cg2,cg3,c△candcj 2 Expansion for the Weierstrass Function 3 Bernoulli Numbers Chapter 5 The Modular Equation 1 Integral Matrices with Positive Determinant 2 The Modular Equation 3 Relations with Isogenies Chapter 6 Higher Levels 1 Congruence Subgroups 2 The Field of Modular Functions OvercC 3 The Field of Modular Functions OvercQ 4 Subfields of the Modular Function Field Chapter 7 Automorphisms of the Modular Function Field 1 Rational Adeles of GL 2 Operation of the Rational Adelescon the Modular Function Field 3 The Shimura Exact Sequence PARTcTWOccCOMPLEXcMULTIPLICATION ELLIPTICcCURVEScWITHcSINGULARcINVARIANTS Chapter 8 Results from Algebraic Number Theory 1 Latticescin Quadratic Fields 2 Completions 3 The Decomposition Group and Frobenius Automorphism 4 Summary of Class Field Theory Chapter 9 Reduction of Elliptic Curves 1 Non-degenerate Reduction, General Case 2 Redu tion of Homomorphisms 3 Coverings of LevelcN 4 Reduction of Differential Forms Chapter 10 Complex Multiplication 1 Generation of Class Fields, Deuring's Approach 2 Idelic Formulation for Arbitrary Lattices 3 Generation of Class Fields by Singular Values of Modular Functions 4 The Frobenius Endomorphism Appendix A Relation of Kronecker Chapter 11 Shimura's Reciprocity Law I Relation Between Generic and Special Extensions 2 Application to Quotientscof Modular Forms Chapter 12 The Fun tion △(at)/△(t) 1 Behavior Under the Artin Automorphism 2 Prime Factorization of its Values 3 Analyti Proof for the Congruence Relationcofj Chapterc13 The l-adic and p-adic Representations of Deuring 1 Thecl-adic Spaces 2 Representations in Characteristi p 3 Representations and Isogenies 4 ReductioncofcthecRingcofcEndomorphisms 5 The Deuring Lifting Theorem Chapter 14 Ihara's Theory 1. Deuring Representatives 2 The Generic Situation 3 Special Situations PART THREE ELLIPTIC CURVEScWITH NON-INTEGRAL INVARIANT Chapter 15 The Tate Parametrization 1 Elliptic Curves with Non-integral Invariants 2 Ellipti Curves Over a Complete Local Ring Chapter 16 The Isogeny Theorems 1 The Galois p-adic Representations 2 Results of Kummer Theory 3 The Local Isogeny Theorems 4 Supersingular Redu tion 5 The Global Isogeny Theorems Chapter 17 Division Points Over Number Fields 1 AcTheorem of Shafarevic 2 The Irreducibility Theorem 3 The Horizontal Galois Group 4 The Vertical Galois Group 5 End of the Proof PARTcFOURccTHETAcFUNCTIONScANDcKRONECKERcLIMIT FORMULA Chapter 18 Product Expansions 1 The Sigma and Zeta Function Appendix The Skew Symmetric Pairing 2 A Normalization and the q-product for the a-function 3 q-expansions Again 4 The q-product forcA 5 The Eta Function of Dedekind 6 Modular Functions of Levelc2 Chapter 19 The Siegel Functions and Klein Forms 1 The Klein Forms 2 The Siegel Functions 3 Special Values of the Siegel Functions Chapter 20 The Kronecker Limit Formulas 1 The Poisson Summation Formula 2 Examples 3 The FunctioncKs(x) 4 The Kronecker First Limit Formula 5 The Kronecker Second LimitcFormula Chapter 21 The First Limit Formula and L-series 1 Relation with L-series 2 The Frobenius Determinant 3 Application to thecL-series Chapter 22 The Second Limit Formula and L-series 1 Gauss Sums 2 An Expression for the L-series APPENDICES ELLIPTIC CURVES IN CHARACTERISTIC p Appendixc1 Algebraic Formulas in Arbitrary Chara teristic BYcJ.cTATE 1 Generalized Weierstrass Form 2 Canonical Forms 3 Expansion Near O; The Formal Group Appendix 2 The Tracecof Frobenius and the Differential of FirstcKind 1 The Trace of Frobenius 2 Duality 3 The Tate Trace 4 The Cartier Operator 5 The Hasse Invariant Bibliography Index
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