Selecta Mathematica

, Volume 23, Issue 2, pp 1205–1234

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

Open Access

DOI: 10.1007/s00029-017-0303-0

Cite this article as:
Hansen, D. & Thorne, J.A. Sel. Math. New Ser. (2017) 23: 1205. doi:10.1007/s00029-017-0303-0


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.

Mathematics Subject Classification

Primary 11F75 Secondary 11F85 
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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA
  2. 2.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK

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