p-adic approaches to the Langlands program

Abstract

We develop and utilize p-adic Hodge theory in families in the context of local-global aspects of the Langlands program.

Our first result allows one to interpolate Hodge-Tate and de Rham periods when some Hodge-Tate-Sen weights are fixed. This is a common generalization of results of Kisin in the case where one weight is fixed and of Berger-Colmez where every weight is fixed. Our main technique is to systematically prove interpolation results for the first cohomology group, where it is possible to obtain base change results regardless of the geometry of the family, and then use algebraic methods to deduce results for periods. We also obtain vanishing for higher cohomology. Varma has applied the main result to show that the Galois representations constructed by Harris-Lan-Taylor-Thorne are de Rham.

Our second result concerns the transfer of regular cuspidal automorphic representations on unitary similitude groups to general linear groups. We build on work of Morel, Shin, and Skinner by proving compatibility at places where the unitary group is ramified. We first obtain compatibility up to monodromy by purely automorphic methods. By applying a crystalline period interpolation result of Kisin and Nakamura to a family constructed using Urban's eigenvariety, we are able to improve this to full compatibility. Two subtleties that arise are (1) the construction of a suitable p-stabilization in the ramified setting, which uses work of Reeder and Lusztig, and (2) the placement of a suitable finite slope representation into a p-adic family, which requires studying the Eisenstein cohomology of the unitary group. We obtain new cases of the generalized Ramanujan conjecture and produce the first examples of strong functorial transfers to generalized linear groups for cuspidal automorphic representations on unitary similitude groups. These strong transfers are used by Skinner-Urban in their work on the Bloch-Kato conjecture.