Borcherds products

Borcherds products are certain meromorphic modular forms on orthogonal
groups whose zeros and poles are supported on Heegner divisors and which have infinite
product expansions analogous to the classical Dedekind eta function. They arise as
regularized theta lifts of weakly holomorphic modular forms.

Borcherds products were discovered by Richard Borcherds in connection with his work on
generalized Kac-Moody algebras and the Moonshine conjectures. Later, researchers
established various connections to other fields of mathematics and mathematical physics.
Borcherds products have found many different applications for instance in Lie theory, algebraic
and arithmetic geometry, number theory, combinatorics, mathematical physics, and
topology.

The mini-course on Borcherds products will give an introduction to the central aspects of
the field with an emphasis on geometric and arithmetic applications.
Topics that will be discussed:

1. Weakly holomorphic modular forms and harmonic Maass forms.
2. Theta functions and liftings.
3. Modular forms on orthogonal groups.
4. Borcherds products and automorphic Green functions.
5. Modularity of the generating series of Heegner divisors.
6. The converse theorem.
7. The Gross-Zagier formula.