Converse theorems and the local Langlands correspondence in families

We prove a descent criterion for certain families of smooth representations of GL_n(F) (F a p-adic field) in terms of the gamma factors of pairs constructed in previous work of the second author. We then use this descent criterion, together with a theory of gamma factors for families of representations of the Weil group W_F (developed previously by both authors), to prove a series of conjectures, due to the first author, that give a complete description of the integral Bernstein center in terms of Galois theory and the local Langlands correspondence. An immediate consequence is the conjectural "local Langlands correspondence in families" of Emerton and Helm.