pdf current version (1.31)

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

1. Preliminaries
2. Elliptic modular curves as Riemann surfaces
3. Elliptic functions
4. Modular functions and modular forms
5. Hecke operators
6. The modular equation for Gamma_0(N)
7. The canonical model of X_0(N) over Q
8. Modular curves as moduli varieties
9. Modular forms, Dirichlet series, and functional equations
10. Correspondences on curves; the theorem of Eichler and Shimura
11. Curves and their zeta functions
12. Complex multiplication for elliptic curves

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.
v1.31; March 22, 2017; corrected; minor revisions; 134 pages.