Elliptic Functions

1. Front Matter

   Pages i-xi

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2. General Theory

   1. Front Matter

      Pages 1-3

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   2. Elliptic Functions

      • Serge Lang

      Pages 5-21

   3. Homomorphisms

      • Serge Lang

      Pages 23-28

   4. The Modular Function

      • Serge Lang

      Pages 29-41

   5. Fourier Expansions

      • Serge Lang

      Pages 43-50

   6. The Modular Equation

      • Serge Lang

      Pages 51-59

   7. Higher Levels

      • Serge Lang

      Pages 61-74

   8. Automorphisms of the Modular Function Field

      • Serge Lang

      Pages 75-84

3. Complex Multiplication Elliptic Curves with Singular Invariants

   1. Front Matter

      Pages 85-87

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   2. Results from Algebraic Number Theory

      • Serge Lang

      Pages 89-109

   3. Reduction of Elliptic Curves

      • Serge Lang

      Pages 111-121

   4. Complex Multiplication

      • Serge Lang

      Pages 123-147

   5. Shimura’s Reciprocity Law

      • Serge Lang

      Pages 149-159

   6. The Function Δ(ατ)/Δ(τ)

      • Serge Lang

      Pages 161-170

   7. The ι-adic and p-adic Representations of Deuring

      • Serge Lang

      Pages 171-186

   8. Ihara’s Theory

      • Serge Lang

      Pages 187-192

4. Elliptic Curves with Non-Integral Invariants

   1. Front Matter

      Pages 193-195

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   2. The Tate Parametrization

      • Serge Lang

      Pages 197-204

   3. The Isogeny Theorems

      • Serge Lang

      Pages 205-220

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.

• Modular form
• complex analysis
• elliptic function
• integral
• operator
• theta function

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

• Department of Mathematics, Yale University, New Haven, USA

  Serge Lang

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