On arithmetic quantum chaos

I will give an introduction to the topic of arithmetic quantum chaos, focusing on the problem of quantum ergodicity. The main question is the following: can the eigenfunctions of the Laplacian, such as automorphic Maaß forms, localise in space as the eigenvalue tends to infinity? In number theory, we may also ask: what happens when the level of the automorphic form tends to infinity? I will present a conjecture made in joint work with Asbjørn Nordentoft, based on a history of conjectures and results by Rudnick--Sarnak, Kowalski—Michel—VanderKam, Nelson, etc., and how out conjecture follows from the Generalised Riemann Hypothesis through a subtle analysis of L-functions.