Elliptic Functions

by Serge Lang

Overview

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.

Product Details

ISBN-13:	9781461291428
Publisher:	Springer New York
Publication date:	09/23/2011
Series:	Graduate Texts in Mathematics , #112
Edition description:	2nd ed. 1987. Softcover reprint of the original 2nd ed. 1987
Pages:	328
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

One General Theory.- 1 Elliptic Functions.- 2 Homomorphisms.- 3 The Modular Function.- 4 Fourier Expansions.- 5 The Modular Equation.- 6 Higher Levels.- 7 Automorphisms of the Modular Function Field.- Two Complex Multiplication Elliptic Curves With Singular Invariants.- 8 Results from Algebraic Number Theory.- 9 Reduction of Elliptic Curves.- 10 Complex Multiplication.- 11 Shimura’s Reciprocity Law.- 12 The Function—(??)/?(?).- 13 The—-adic and p-adic Representations of Deuring.- 14 Ihara’s Theory.- Three Elliptic Curves with Non-Integral Invariant.- 15 The Tate Parametrization.- 16 The Isogeny Theorems.- 17 Division Points Over Number Fields.- Four Theta Functions and Kronecker Limit Formula.- 18 Product Expansions.- 19 The Siegel Functions and Klein Forms.- 20 The Kronecker Limit Formulas.- 21 The First Limit Formula and L-series.- 22 The Second Limit Formula and 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.

Editorial Reviews

Second Edition

S. Lang

Elliptic Functions

"This book is an excellent addition to the literature and provides a rather complete picture of the modern theory of elliptic functions while remaining at a very concrete level."—MATHEMATICAL REVIEWS

From the Publisher