Elliptic Functions

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. The book is divided into four parts. In the first, Lang presents the general analytic theory starting from scratch. Most of this can be read by a student with a basic knowledge of complex analysis. The next part treats complex multiplication, including a discussion of Deuring's theory of l-adic and p-adic representations, and elliptic curves with singular invariants. Part three covers curves with non-integral invariants, and applies the Tate parametrization to give Serre's results on division points. The last part covers theta functions and the Kronecker Limit Formula. Also included is an appendix by Tate on algebraic formulas in arbitrary charactistic.

One General Theory.- 1 Elliptic Functions.- 1 The Liouville Theorems.- 2 The Weierstrass Function.- 3 The Addition Theorem.- 4 Isomorphism Classes of Elliptic Curves.- 5 Endomorphisms and Automorphisms.- 2 Homomorphisms.- 1 Points of Finite Order.- 2 Isogenies.- 3 The Involution.- 3 The Modular Function.- 1 The Modular Group.- 2 Automorphic Functions of Degree 2k.- 3 The Modular Function j.- 4 Fourier Expansions.- 1 Expansion for Gk, g2, g3, ? and j.- 2 Expansion for the Weierstrass Function.- 3 Bernoulli Numbers.- 5 The Modular Equation.- 1 Integral Matrices with Positive Determinant.- 2 The Modular Equation.- 3 Relations with Isogenies.- 6 Higher Levels.- 1 Congruence Subgroups.- 2 The Field of Modular Functions Over C.- 3 The Field of Modular Functions Over Q.- 4 Subfields of the Modular Function Field.- 7 Automorphisms of the Modular Function Field.- 1 Rational Adeles of GL2.- 2 Operation of the Rational Adeles on the Modular Function Field.- 3 The Shimura Exact Sequence.- Two Complex Multiplication Elliptic Curves With Singular Invariants.- 8 Results from Algebraic Number Theory.- 1 Lattices in Quadratic Fields.- 2 Completions.- 3 The Decomposition Group and Frobenius Automorphism.- 4 Summary of Class Field Theory.- 9 Reduction of Elliptic Curves.- 1 Non-degenerate Reduction, General Case.- 2 Reduction of Homomorsphisms.- 3 Coverings of Level N.- 4 Reduction of Differential Forms.- 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.- 11 Shimura’s Reciprocity Law.- 1 Relation Between Generic and Special Extensions.- 2 Application to Quotients of Modular Forms.- 12 The Function ?(??)/?(?).- 1 Behavior Under the Artin Automorphism.- 2 Prime Factorization of its Values.- 3 Analytic Proof for the Congruence Relation of j.- 13 The ?-adic and p-adic Representations of Deuring.- 1 The ?-adic Spaces.- 2 Representations in Characteristic p.- 3 Representations and Isogenies.- 4 Reduction of the Ring of Endomorphisms.- 5 The Deuring Lifting Theorem.- 14 Ihara’s Theory.- 1 Deuring Representatives.- 2 The Generic Situation.- 3 Special Situations.- Three Elliptic Curves with Non-Integral Invariant.- 15 The Tate Parametrization.- 1 Elliptic Curves with Non-integral Invariants.- 2 Elliptic Curves Over a Complete Local Ring.- 16 The Isogeny Theorems.- 1 The Galois p-adic Representations.- 2 Results of Kummer Theory.- 3 The Local Isogeny Theorems.- 4 Supersingular Reduction.- 5 The Global Isogeny Theorems.- 17 Division Points Over Number Fields.- 1 A Theorem of Shafarevi?.- 2 The Irreducibility Theorem.- 3 The Horizontal Galois Group.- 4 The Vertical Galois Group.- 5 End of the Proof.- Four Theta Functions and Kronecker Limit Formula.- 18 Product Expansions.- 1 The Sigma and Zeta Function.- Appendix The Skew Symmetric Pairing.- 2 A Normalization and the q-product for the ?-function.- 3 q-expansions Again.- 4 The q-product for ?.- 5 The Eta Function of Dedekind.- 6 Modular Functions of Level 2.- 19 The Siegel Functions and Klein Forms.- 1 The Klein Forms.- 2 The Siegel Functions.- 3 Special Values of the Siegel Functions.- 20 The Kronecker Limit Formulas.- 1 The Poisson Summation Formula.- 2 Examples.- 3 The Function Ks(x).- 4 The Kronecker First Limit Formula.- 5 The Kronecker Second Limit Formula.- 21 The First Limit Formula and L-series.- 1 Relation with L-series.- 2 The Frobenius Determinant.- 3 Application to the L-series.- 22 The Second Limit Formula and L-series.- 1 Gauss Sums.- 2 An Expression for the L-series.- Appendix 1 Algebraic Formulas in Arbitrary Characteristic.- By J. Tate.- 1 Generalized Weierstrass Form.- 2 Canonical Forms.- Appendix 2 The Trace of Frobenius and the Differential of First Kind.- 1 The Trace of Frobenius.- 2 Duality.- 3 The Tate Trace.- 4 The Cartier Operator.- 5 The Hasse Invariant.