Selecta Mathematica

, Volume 23, Issue 2, pp 1205–1234 | Cite as

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

  • David Hansen
  • Jack A. ThorneEmail author
Open Access


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 



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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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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