The Cohomological Sarnak-Xue Density Hypothesis for $SO_5$
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2023
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| author | Evra, Shai Gerbelli-Gauthier, Mathilde Gustafsson, Henrik P. A. |
| author_facet | Evra, Shai Gerbelli-Gauthier, Mathilde Gustafsson, Henrik P. A. |
| contents | 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 Taïbi 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_12413 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Cohomological Sarnak-Xue Density Hypothesis for $SO_5$ Evra, Shai Gerbelli-Gauthier, Mathilde Gustafsson, Henrik P. A. Number Theory Representation Theory 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 Taïbi 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. |
| title | The Cohomological Sarnak-Xue Density Hypothesis for $SO_5$ |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2309.12413 |