$L^p$-norm bounds for automorphic forms

A fundamental problem in analysis is understanding the distribution of mass of Laplacian eigenfunctions via bounds for their $L^p$ norms in terms of the size of their Laplacian eigenvalue. Number theorists are interested in the Laplacian eigenfunctions on the modular surface that are additionally joint eigenfunctions of every Hecke operator---namely the Hecke--Maass cusp forms. In this talk, I will describe joint work with Peter Humphries in which we prove new bounds for $L^p$ norms in this situation. This is achieved by using $L$-functions and their reciprocity formulae: certain special identities between two different moments of central values of $L$-functions.