p-adic Langlands functoriality

Abstract

In this thesis we prove p-adic Langlands functoriality in two settings. Firstly we prove a p-adic Labesse-Langlands transfer from the group of units in a defnite quaternion algebra to its subgroup of norm one elements. Secondly we show p-adic functoriality for inner forms of unitary groups in three variables. In both cases we show that given an eigenvariety for the first group, there exists an eigenvariety for the second group and a morphism between them that extends the classical Langlands transfer and has an interpretation as a p-adic Langlands transfer.