Sums of Kloosterman sums and sums of L-functions

This talk is a review a recent series of works by V. Blomer, E. Fouvry, E. Kowalski,
myself, D. Milicevic, W. Sawin as well as  R. Zacharias.

We will describe various estimates on sums of Kloosterman sums (or more generally
trace functions) proven using methods from $\ell$-adic cohomology  and some of  their
applications  to the study of analytic properties of  character twists of L-functions on
average over the family of Dirichlet characters of some large prime modulus.