Low moments of Hecke eigenvalue sums

Abstract

We show that partial sums of the Sato--Tate random multiplicative functions introduced by Cogdell and Michel exhibit better-than-square-root cancellation. The proof proceeds via a connection to multiplicative chaos, following Harper's seminal work. By a non-trivial adaptation of Harper's derandomization argument for character sums, we also obtain upper bounds for low moments of Hecke eigenvalue sums and of Hecke eigenforms near the cusp $0$; to our knowledge, this is the first appearance of multiplicative chaos in the context of automorphic forms on $\mathrm{GL}(2)$. A novel ingredient is the introduction of Hecke $s$-norms.

42 pages