Joint IAS/Princeton University Number Theory Seminar

In joint work with Emmanuel Kowalski and Philippe Michel, we prove two different estimates on sums of coefficients of modular forms---one related to L-functions and another to the level of distribution. A key step in the argument is a careful analysis of vanishing cycles, a tool originally developed by Lefschetz to study the topology of algebraic varieties. We will explain why this is helpful for these problems.