Joint PU/IAS Number Theory

Central predictions of arithmetic quantum chaos such as the Quantum Unique Ergodicity conjecture and the sup-norm problem ask about the mass distribution of automorphic forms, most classically in terms of their weight or Laplace eigenvalue (for classical and Maass forms, respectively), but also in terms of the volume of the manifold and other parameters. The full automorphic spectrum on a reductive group G involves forms of all possible types under its maximal compact subgroup K, and it is natural to ask about the impact of the non-spherical spectrum (in particular, in the large dimension limit) on the mass concentration and spectral geometry on G.

In this walk, we will motivate the sup-norm problem and then describe our results, joint with Blomer, Harcos, and Maga, solving the problem for non-spherical Maass forms of an increasing dimension of the associated K-type in two settings, on arithmetic quotients of G=SL(2,C) and G=SL(3,R). We describe in detail the mass concentration of spherical trace functions of arbitrary K-type and are led to new counting problems of Hecke correspondences close to various special submanifolds of G.