Sean Howe ( U. Chicago)
Overconvergent modular forms and the p-adic Jacquet-Langlands correspondence.

Abstract:We explain an explicit transfer of Hecke eigensystems from the space of overconvergent modular forms to the space of continuous p-adic automorphic functions on the units of the definite quaternion algebra of invariant p, giving a partial answer to an old question of Serre. Conjecturally, these p-adic automorphic functions should satisfy a local-global compatibility with the local p-adic Jacquet-Langlands of Knight and Scholze; if this holds, then our construction can be used to obtain new information about the quaternion algebra representations arising in this correspondence. The construction proceeds by evaluating overconvergent modular forms at special points in the infinite level modular curve. To make sense of this evaluation we employ a construction of overconvergent modular forms using the infinite level modular curve and the Hodge-Tate period map. Control over the quaternion algebra representation and the field of coefficients is obtained from a reciprocity law intertwining the Galois, GL_2, and quaternion algebra actions.