Overconvergent Modular Forms and the p-Adic Jacquet-Langlands Correspondence

We construct a global p-adic Jacquet-Langlands transfer from overconvergent modular forms,to naive p-adic automorphic forms on the quaternion algebra over Q ramified at p and,infinity, answering an old question of Serre [26, paragraph (26)]. Using this transfer, we show,that the completed Hecke algebra of naive automorphic forms on the quaternion algebra is,isomorphic to the completed Hecke algebra of modular forms, and, conditional on a local-global,compatibility conjecture, obtain new information about the local p-adic Jacquet-,Langlands correspondence of Knight and Scholze. The construction and proofs live entirely,in the world of p-adic geometry; in particular we do not use the smooth Jacquet-Langlands,correspondence as an input.