Derived Hecke action on the trivial cohomology of division algebras
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916881287348224 |
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| author | Sahu, Soumyadip |
| author_facet | Sahu, Soumyadip |
| contents | This article generalizes Venkatesh's structure theorem for the derived Hecke action on the Hecke trivial cohomology of a division algebra over an imaginary quadratic field to division algebras over all number fields. In particular, we show that the stable submodule of the Hecke trivial cohomology attached to a division algebra is a free module generated by the unit class for the action of the strict derived Hecke algebra. Moreover, the strict derived Hecke algebra possesses a rational form that preserves the canonical rational structure on the stable cohomology during the derived Hecke action. The main ingredients in our improvement are a careful study of the congruence classes in the torsion cohomology of the arithmetic manifold and the author's new result on the reduction map in the $K$-theory of the ring of integers in number fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06732 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Derived Hecke action on the trivial cohomology of division algebras Sahu, Soumyadip Number Theory This article generalizes Venkatesh's structure theorem for the derived Hecke action on the Hecke trivial cohomology of a division algebra over an imaginary quadratic field to division algebras over all number fields. In particular, we show that the stable submodule of the Hecke trivial cohomology attached to a division algebra is a free module generated by the unit class for the action of the strict derived Hecke algebra. Moreover, the strict derived Hecke algebra possesses a rational form that preserves the canonical rational structure on the stable cohomology during the derived Hecke action. The main ingredients in our improvement are a careful study of the congruence classes in the torsion cohomology of the arithmetic manifold and the author's new result on the reduction map in the $K$-theory of the ring of integers in number fields. |
| title | Derived Hecke action on the trivial cohomology of division algebras |
| topic | Number Theory |
| url | https://arxiv.org/abs/2501.06732 |