in a 1987 letter to tate, serre showed that the hecke eigensystems appearing in mod p modular forms are the same as those appearing in mod p functions on a finite double coset constructed from the quaternion algebra ramified at p and infinity. at the end of the letter, he asked whether there might be a similar relation between p-adic modular forms and p-adic functions on the quaternion algebra. we show the answer is yes: the completed hecke algebra of p-adic modular forms is the same as the completed hecke algebra of naive p-adic automorphic functions on the quaternion algebra. the resulting p-adic jacquet-langlands correspondence is richer than the classical jacquet-langlands correspondence -- for example, ramanujan's delta function, which is invisible to the classical correspondence, appears. the proof is a lifting of serre's geometric argument from characteristic p to characteristic zero; the quaternionic double coset is realized as a fiber of the hodge-tate period map, and eigensystems are extended off of the fiber using a variant of scholze's fake hasse invariants.