| 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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