Semistable periods of finite slope families

Abstract

We introduce the notion of finite slope families to encode the local properties of the $p$-adic families of Galois representations appearing in the work of Harris, Lan, Taylor and Thorne on the construction of Galois representations for (non-self-dual) regular algebraic cuspidal automorphic representations of $GL\left(n\right)$ over CM fields. Our main result is to prove the analytic continuation of semistable (and crystalline) periods for such families.