Elliptic Functions

Serge Lang (auth.)9780387965086, 0387965084

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.

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Table of contents :
Front Matter….Pages i-xi
Front Matter….Pages 1-3
Elliptic Functions….Pages 5-21
Homomorphisms….Pages 23-28
The Modular Function….Pages 29-41
Fourier Expansions….Pages 43-50
The Modular Equation….Pages 51-59
Higher Levels….Pages 61-74
Automorphisms of the Modular Function Field….Pages 75-84
Front Matter….Pages 85-87
Results from Algebraic Number Theory….Pages 89-109
Reduction of Elliptic Curves….Pages 111-121
Complex Multiplication….Pages 123-147
Shimura’s Reciprocity Law….Pages 149-159
The Function Δ(ατ)/Δ(τ)….Pages 161-170
The ι-adic and p-adic Representations of Deuring….Pages 171-186
Ihara’s Theory….Pages 187-192
Front Matter….Pages 193-195
The Tate Parametrization….Pages 197-204
The Isogeny Theorems….Pages 205-220
Division Points over Number Fields….Pages 221-233
Front Matter….Pages 235-237
Product Expansions….Pages 239-257
The Siegel Functions and Klein Forms….Pages 259-266
The Kronecker Limit Formulas….Pages 267-278
Front Matter….Pages 235-237
The First Limit Formula and L-series….Pages 279-285
The Second Limit Formula and L-series….Pages 287-293
Back Matter….Pages 295-328