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arXiv · 2309.12413

The Cohomological Sarnak-Xue Density Hypothesis for $SO_5$

Shai Evra·Mathilde Gerbelli-Gauthier·Henrik P. A. Gustafsson
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Abstract

We prove the cohomological version of the Sarnak--Xue Density Hypothesis for $SO_{5}$ over a totally real field and for inner forms split at all finite places. The proof relies on recent lines of work in the Langlands program: (i) Arthur's Endoscopic Classification of Representations of classical groups, extended to inner forms by Ta\"ibi and its explicit description for $SO_{5}$ by Schmidt, and (ii) the Generalized Ramanujan--Petersson Theorem, proved for cohomological self-dual cuspidal representations of general linear groups. We give applications to the growth of cohomology of arithmetic manifolds, density-Ramanujan complexes, cutoff phenomena and optimal strong approximation.

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Shai Evra, Mathilde Gerbelli-Gauthier, Henrik P. A. Gustafsson. 2023-09-21. The Cohomological Sarnak-Xue Density Hypothesis for $SO_5$. https://arxiv.org/abs/2309.12413

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