Joint IAS/Princeton University Number Theory Seminar

Given an L^2-normalized cusp form f on a modular curve X_0(N) , what can be said about pointwise bounds for f ? For Hecke eigenforms, we will prove the first non-trivial bound in terms of the level N as well as hybrid bounds in terms of the level and the Laplacian eigenvalue. Similar techniques work for functions on other spaces, e.g. quotients of quaternion algebras. This is joint work with R. Holowinsky.