The random wave conjecture and arithmetic quantum chaos

Berry's random wave conjecture is a heuristic that the eigenfunctions of a classically ergodic system ought to display Gaussian random behaviour, as though they were random waves, in the large eigenvalue limit.

We discuss two aspects of this problems for eigenfunctions of the Laplacian on a particular number-theoretic negatively curved surface: Planck scale mass equidistribution, and an asymptotic for the fourth moment.

We will highlight how the resolution of these two problems in this number-theoretic setting involves a delicate understanding of the behaviour of certain families of L-functions.