A p-adic Jacquet-Langlands Correspondence

Abstract

The Jacquet-Langlands correspondence asserts that an automorphic representation of GL(2) transfers to an automorphic representation of a quaternion algebra over Q if and only if its local component is square-integrable at the places that are ramified in the quaternion algebra. It is known that the local representation of GL(2, R) associated to a cusp form of weight one is not square-integrable. Thus, weight 1 forms do not transfer to a quaternion algebra ramified at infinity. Nevertheless, fixing a prime p split in the quaternion algebra that is ramified at infty, we will discuss a p-adic formulation of the Jacquet-Langlands correspondence that includes cuspidal newforms of weight 1 that are supercuspidal at p.