Seminar

This is a report on joint work in progress with Darmon, Rotger, and Venkatesh.    In a series of papers, Venkatesh and his collaborators have proposed a conjectural framework that incorporates the action of motivic cohomology groups on the cohomology of locally symmetric spaces.  The simplest case concerns the cohomology of the special fiber in characteristic p of the modular curve with coefficients in the sheaf of weight 1 forms. There the conjecture relates the action of derived Hecke operators, which only exist over coefficient rings in positive characteristic, to the (discrete) logarithms of certain units in the number fields cut out by two-dimensional complex representations of the absolute Galois group of Q.  Our main result is a proof of this conjecture as it pertains to weight one forms whose associated Galois representations are induced from Dirichlet characters of quadratic fields.  The proof is based on classical constructions in the theory of modular forms. If time permits, there will be a discussion of how Venkatesh's conjectures might fit into a (highly speculative) derived version of the Langlands program.