On the GLₙ GL n -eigenvariety and a conjecture of Venkatesh

Let π be a cuspidal, cohomological automorphic representation of GLₙ( A) . Venkatesh has suggested that there should exist a natural action of the exterior algebra of a certain motivic cohomology group on the π -part of the Betti cohomology (with rational coefficients) of the GLₙ( 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.