Sato–Tate equidistribution for families of Hecke–Maass forms on SL(n, ℝ)∕SO(n)

Matz, Jasmin; Templier, Nicolas

doi:10.2140/ant.2021.15.1343

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Sato–Tate equidistribution for families of Hecke–Maass forms on $\mathrm{SL}(n,\mathbb{R}) / \mathrm{SO}(n)$

Jasmin Matz and Nicolas Templier

Vol. 15 (2021), No. 6, 1343–1428

DOI: 10.2140/ant.2021.15.1343

Abstract

We establish the Sato–Tate equidistribution of Hecke eigenvalues of the family of Hecke–Maass cusp forms on $SL (n, ℤ) ∖ SL (n, ℝ) ∕ SO (n)$ . As part of the proof, we establish a uniform upper-bound for spherical functions on semisimple Lie groups which is of independent interest. For each of the principal, symmetric square and exterior square $L$ -functions, we deduce the level distribution with restricted support of the low-lying zeros. We also deduce average estimates toward Ramanujan, including an improvement on the previous literature in the case $n = 2$ .

Keywords

automorphic forms, $L$-functions, Arthur–Selberg trace formula

Mathematical Subject Classification

Primary: 11F70

Secondary: 20G30, 33C55, 43A90

Milestones

Received: 1 September 2018

Revised: 24 November 2020

Accepted: 20 December 2020

Published: 16 October 2021

Authors

Jasmin Matz
Department of Mathematical Science Universitetsparken 5 Copenhagen Denmark

Nicolas Templier
Department of Mathematics Cornell University Ithaca, NY United States

Vol. 15, No. 6, 2021