General Prerequisites: Part A Number Theory, Topology and Part B Geometry of Surfaces, Algebraic Curves (or courses covering similar material) are useful but not essential.

Course 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.

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 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.

Course Synopsis: Overview and examples of modular forms. Definition and basic properties of modular forms.

Topology of modular curves: a fundamental domain for the full modular group; fundamental domains for subgroups \(\Gamma\) of finite index in the modular group; the compact surfaces X\(\Gamma\); explicit triangulations of X\(\Gamma\) and the computation of the genus using the Euler characteristic formula; the congruence subgroups \(\Gamma\)(N),\(\Gamma\)0(N) and \(\Gamma\)1(N); examples of genus computations.

Dimensions of spaces of modular forms: general dimension formula (proof non-examinable); the valence formula (proof non-examinable).

Examples of modular forms: Eisenstein series in level 1; Ramanujan's \(\Delta\)-function; some arithmetic applications.

The Petersson inner product.

Modular forms as functions on lattices: modular forms of level 1 as functions on lattices; Eisenstein series revisited.

Hecke operators in level 1: Hecke operators on lattices; Hecke operators on modular forms and their q-expansions; Hecke operators are Hermitian; multiplicity one.