Synopsis for Modular Forms

Prerequisites

Part A Analysis and Algebra (core material) and Part A Group Theory. Part A Number Theory is useful but not essential. B3b Algebraic Curves is recommended but some background reading on the notions of a Riemann surface and its genus will suffice.

Overview

The course aims to introduce students to the beautiful theory of modular forms, one of the cornerstones of modern number theory. This theory is a rich and challenging blend of methods from complex analysis and linear algebra, and an explicit application of group actions and the theory of Riemann surfaces.

Learning Outcomes

The student will learn about modular curves and spaces of modular forms, and understand in special cases how to compute their genus and dimension, respectively. They will see how modular curves parameterise families of elliptic curves, and that modular forms can be described explicitly via their q-expansions, and they will be familiar with explicit examples of modular forms. They will learn about the rich algebraic structure on spaces of modular forms, given by Hecke operators and the Petersson inner product, how spaces of modular forms of different level are related, and how modular forms may be used in number theory.

Synopsis

The modular group and the upper half-plane. Lattices and elliptic curves. Modular forms of level 1. Examples of modular forms: Eisenstein series, Ramanujan's function Congruence subgroups and fundamental domains. Modular forms of higher level. Hecke operators. The Petersson inner product. Old and new forms. Applications in number theory.

1. F. Diamond and J. Shurman, A First Course in Modular Forms, Graduate Texts in Mathematics 228, (Parts of Chapters 1-5), Springer-Verlag, 2005.
2. J.-P. Serre, A Course in Arithmetic, (Chapter VII), Graduate Texts in Mathematics 7, Springer-Verlag, 1973.