Overconvergence of \'etale $(\varphi,\Gamma)$-modules in families

TL;DR

This paper proves a conjecture about the overconvergence property of étale $(, abla)$-modules in families, establishing a link between algebraic and analytic stacks in $p$-adic Hodge theory.

Contribution

It confirms a conjecture on overconvergence in families of étale $(, abla)$-modules and constructs a natural map between associated stacks.

Findings

Proves overconvergence conjecture for étale $(, abla)$-modules in families.

Establishes a natural map from the Emerton-Gee stack to the stack of $(, abla)$-modules.

Bridges algebraic and analytic perspectives in $p$-adic Hodge theory.

Abstract

We prove a conjecture of Emerton, Gee and Hellmann concerning the overconvergence of \'etale $(\varphi,\Gamma)$-modules in families parametrized by topologically finite type $\mathbb{Z}_{p}$-algebras. As a consequence, we deduce the existence of a natural map from the rigid fiber of the Emerton-Gee stack to the rigid analytic stack of $(\varphi,\Gamma)$-modules.