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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 |
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Vol. 15 (2021), No. 6, 1343–1428
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Abstract |
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We establish the Sato–Tate equidistribution of Hecke eigenvalues of the family of Hecke–Maass cusp forms on . 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 -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 . |
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Keywords
automorphic forms, $L$-functions, Arthur–Selberg trace
formula
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Mathematical Subject Classification
Primary: 11F70
Secondary: 20G30, 33C55, 43A90
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Milestones
Received: 1 September 2018
Revised: 24 November 2020
Accepted: 20 December 2020
Published: 16 October 2021
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Authors |
| Jasmin Matz | |
| Department of Mathematical
Science Universitetsparken 5 Copenhagen Denmark |
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| Nicolas Templier | |
| Department of Mathematics Cornell University Ithaca, NY United States |
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