Modular Functions and Modular Forms
J. S. Milne
Abstract. These are the notes for Math 678, University of Michigan, Fall 1990, exactly as they were handed out during the course except for some minor revisions andcorrections. Please sendcomments andcorrections to me at [email protected].
Contents
Introduction 1 Riemann surfaces 1 The general problem 1 Riemann surfaces that are quotients of D.1 Modular functions. 2 Modular forms. 3 Plane affine algebraic curves 3 Plane projective curves. 4 Arithmetic of Modular Curves. 5 Elliptic curves. 5 Elliptic functions. 5 Elliptic curves and modular curves. 6 Books 7 1. Preliminaries 8 Continuous group actions. 8 Riemann surfaces: classical approach 10 Riemann surfaces as ringed spaces 11 Differential forms. 12 Analysis on compact Riemann surfaces. 13 Riemann-Roch Theorem. 15 The genus of X.17 Riemann surfaces as algebraic curves. 18 2. Elliptic Modular Curves as Riemann Surfaces 19