Modular Functions and Modular Forms - J.S. Milne Top

Group Theory

Fields and Galois Theory

Algebraic Geometry

Algebraic Number Theory

Modular Functions and Modular Forms

Elliptic Curves -- see books.

Abelian Varieties

Lectures on Etale Cohomology

Class Field Theory

Complex Multiplication

Basic Theory of Affine Group Schemes

Lie Algebras, Algebraic Groups, and Lie Groups

Reductive Groups

Errata

pdf current version (1.30)
### Abstract

This is an introduction to the
arithmetic theory of modular functions and modular forms, with a greater
emphasis on the geometry than most accounts.
### Contents

### Prerequisites

The algebra and complex analysis usually covered in advanced undergraduate or
first-year graduate courses.
### History

v1.10; May 22, 1997; first version on the web; 128 pages.
old version (1.10)

v1.20; November 23, 2009; new style; minor fixes and improvements; addedlist of symbols; 129 pages.

v1.30; April 26, 2012; corrected; many minor revisions; 138 pages.

- Preliminaries
- Elliptic modular curves as Riemann surfaces
- Elliptic functions
- Modular functions and modular forms
- Hecke operators
- The modular equation for Gamma_0(N)
- The canonical model of X_0(N) over Q
- Modular curves as moduli varieties
- Modular forms, Dirichlet series, and functional equations
- Correspondences on curves; the theorem of Eichler and Shimura
- Curves and their zeta functions
- Complex multiplication for elliptic curves

v1.20; November 23, 2009; new style; minor fixes and improvements; addedlist of symbols; 129 pages.

v1.30; April 26, 2012; corrected; many minor revisions; 138 pages.