Mass equidistribution of Hilbert modular eigenforms

TL;DR

This paper proves that the mass of Hilbert modular eigenforms becomes uniformly distributed on the Hilbert modular variety as their weights increase, extending previous results from the rational case to general totally real fields.

Contribution

It generalizes Holowinsky and Soundararajan's method to totally real fields, establishing mass equidistribution for Hilbert modular eigenforms.

Findings

Mass equidistribution proven for Hilbert modular eigenforms over totally real fields.

Extends the proof technique from rational to general totally real fields.

Addresses the adaptation of bounds for Weyl periods to number fields with units.

Abstract

Let F be a totally real number field, and let f traverse a sequence of non-dihedral holomorphic eigencuspforms on GL(2)/F of weight (k_1,...,k_n), trivial central character and full level. We show that the mass of f equidistributes on the Hilbert modular variety as max(k_1,...,k_n) tends to infinity. Our result answers affirmatively a natural analogue of a conjecture of Rudnick and Sarnak (1994). Our proof generalizes the argument of Holowinsky-Soundararajan (2008) who established the case F = Q. The essential difficulty in doing so is to adapt Holowinsky's bounds for the Weyl periods of the equidistribution problem in terms of manageable shifted convolution sums of Fourier coefficients to the case of a number field with nontrivial unit group.