**Joint Princeton University and IAS Number Theory Seminar **

**2/9/2012**

**Nicolas Templier
Princeton University**

**Sup-norms, Whittaker periods and hypergeometric sums**

We begin with a survey of recent results on the problem of bounding the sup-norm of automorphic forms. If f is a cuspidal automorphic forms on a reductive group G it is classical to study its value distribution and in the particular the maximum of |f(g)| for all g in G. Then we will explain an approach to this problem via Whittaker periods. We establish a new formula for non-archimedean Whittaker functions. The formula involves 2F1 hypergeometric sums and generalizes classical results of Casselman and others. As an application we disprove a folklore conjecture on the sup-norm of GL(2) modular forms.