Converse theorems and the local Langlands correspondence in families

We prove a descent criterion for certain families of smooth representations of [Formula: see text] (F a p-adic field) in terms of the [Formula: see text] -factors of pairs constructed in Moss (Int Math Res Not 2016(16):4903–4936, 2016). We then use this descent criterion, together with a theory of [Formula: see text] -factors for families of representations of the Weil group [Formula: see text] (Helm and Moss in Deligne–Langlands gamma factors in families, arXiv:1510.08743v3, 2015), to prove a series of conjectures, due to the first author, that give a complete description of the center of the category of smooth [Formula: see text] -modules (the so-called “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 (Ann Sci Éc Norm Supér (4) 47(4):655–722, 2014).