On the \(\mathrm {GL}_n\)-eigenvariety and a conjecture of Venkatesh

Abstract

Let \(\pi \) be a cuspidal, cohomological automorphic representation of \(\mathrm {GL}_n({\mathbb A})\). Venkatesh has suggested that there should exist a natural action of the exterior algebra of a certain motivic cohomology group on the \(\pi \)-part of the Betti cohomology (with rational coefficients) of the \(\mathrm {GL}_n({\mathbb Q})\)-arithmetic locally symmetric space. Venkatesh has given evidence for this conjecture by showing that its ‘l-adic realization’ is a consequence of the Taylor–Wiles formalism. We show that its ‘p-adic realization’ is related to the properties of eigenvarieties.

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Acknowledgements

D.H. is grateful to Columbia University’s Faculty Research Allowance Program for funding a trip to the University of Cambridge, during which some progress on this work occurred. In the period during which this research was conducted, J.T. served as a Clay Research Fellow. We thank the referee for helpful remarks.

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Correspondence to Jack A. Thorne.

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Hansen, D., Thorne, J.A. On the \(\mathrm {GL}_n\)-eigenvariety and a conjecture of Venkatesh. Sel. Math. New Ser. 23, 1205–1234 (2017). https://doi.org/10.1007/s00029-017-0303-0

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  • Secondary 11F85