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DERIVED HECKE ALGEBRA AND COHOMOLOGY OF ARITHMETIC GROUPS

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  23 December 2019

AKSHAY VENKATESH
AKSHAY VENKATESH*
Affiliation:
Institute for Advanced Study, Princeton, NJ 08540; akshay@ias.edu

Abstract

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We describe a graded extension of the usual Hecke algebra: it acts in a graded fashion on the cohomology of an arithmetic group $\unicode[STIX]{x1D6E4}$. Under favorable conditions, the cohomology is freely generated in a single degree over this graded Hecke algebra.

From this construction we extract an action of certain $p$-adic Galois cohomology groups on $H^{\ast }(\unicode[STIX]{x1D6E4},\mathbf{Q}_{p})$, and formulate the central conjecture: the motivic $\mathbf{Q}$-lattice inside these Galois cohomology groups preserves $H^{\ast }(\unicode[STIX]{x1D6E4},\mathbf{Q})$.

MSC classification

Primary: 11F75: Cohomology of arithmetic groups 11F80: Galois representations
Type
Research Article
Information
Forum of Mathematics, Pi , Volume 7 , 2019 , e7
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2019

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