We show that the completed Hecke algebra of $p$ -adic modular forms is isomorphic to continuous automorphic for units quaternion ramified at and $\infty$ . This gives an affirmative answer a question posed by Serre in 1987 letter Tate. The proof geometric, lifts mod argument due Serre: we evaluate identifying quaternionic double-coset with fiber Hodge–Tate period map, extend functions off using...