Playlist - Modular forms (Borcherds)

We introduce modular forms, and give several examples of how they were used to solve problems in apparently unrelated areas of mathematics.

We give two ways of looking at modular forms: as functions of lattices in ℂ, or as invariant forms. We use this to give two different ways of constructing Eisenstein series.

We calculate the Fourier coefficients of the Eisenstein series introduced in the previous lecture, and use them to construct the elliptic modular function.

We describe the fundamental domain of SL₂(ℤ) acting on the upper half plane.

We first show that the number of zeros of a (level 1 holomorphic) modular form in a fundamental domain is weight/12, and use this to show that the graded ring of modular forms is the ring of polynomials in E₄ and E₆.

We classify all meromorphic modular functions, showing that they are all rational functions of the elliptic modular function j.

As an application of j we use it to prove Picard's theorem that a non-constant meromorphic function can omit at most one value.

We discuss the infinite product of the discriminant function and relate it to the fact that the Eisenstein series E₂ is not quite a modular form. We then sketch Siegel's proof of the infinite product for the discriminant.

We show that the theta function of a 1-dimensional lattice is a modular form using the Poisson summation formula, and use this to prove the functional equation of the Riemann zeta function.

We study theta functions of even unimodular lattices, such as the root lattice of the E₈ exceptional Lie algebra. As examples we show that one cannot "hear the shape of a drum", and calculate the number of minimal vectors in the Leech lattice, and show that any even unimodular lattice has to have dimension divisible by 8.

We introduce Hecke operators for modular functions in three different ways.

We show how to use Hecke operators to prove the product formula for j (which is the the Weyl denominator function of the monster Lie algebra).

We extend Hecke operators from modular functions to modular forms. As an application we prove some of Ramanujan's conjectures for the Ramanujan τ-function.

We define the Petersson inner product on modular forms and use it to show that the eigenforms of the Hecke algebra span the space of modular forms.