Perfectoidness via Sen Theory and Applications to Shimura Varieties

TL;DR

This paper links Sen's theorem and $p$-adic Hodge theory to establish perfectoidness criteria for Galois extensions and Shimura varieties, leading to new results on their cohomology and confirming a conjecture.

Contribution

It develops a geometric analogue of Sen's criterion for perfectoidness in $p$-adic geometry and applies it to prove vanishing of higher cohomology in Shimura varieties.

Findings

Established a geometric analogue of Sen's criterion for $p$-adic Galois extensions.

Proved the perfectoidness of stalks of Shimura varieties at infinite level.

Verified the vanishing of higher-degree cohomology groups in Shimura varieties.

Abstract

Sen's theorem on the ramification of a $p$-adic analytic Galois extension of $p$-adic local fields shows that its perfectoidness is equivalent to the non-vanishing of its arithmetic Sen operator. By developing $p$-adic Hodge theory for general valuation rings, we establish a geometric analogue of Sen's criterion for any $p$-adic analytic Galois extension of $p$-adic varieties: its (Riemann-Zariski) stalkwise perfectoidness is necessary for the non-vanishing of the geometric Sen operators. As the latter is verified for general Shimura varieties by Pan and Camargo, we obtain the perfectoidness of every completed stalk of general Shimura varieties at infinite level at $p$. As an application, we prove that the integral completed cohomology groups vanish in higher degrees, verifying a conjecture of Calegari-Emerton for general Shimura varieties.