Congruences of first syntomic cohomology groups

TL;DR

This paper establishes a precise criterion for when the mod p^n reductions of first syntomic cohomology groups of reflexive F-gauges are isomorphic, linking it to isomorphisms of associated Breuil--Kisin modules with G_K-actions.

Contribution

It provides a new congruence criterion connecting syntomic cohomology groups and Breuil--Kisin modules in p-adic Hodge theory.

Findings

Isomorphism of mod p^n reductions corresponds to isomorphism of Breuil--Kisin modules mod p^{2n}.

Refines understanding of local Bloch--Kato Selmer groups.

Establishes a congruence relation in p-adic Hodge theory.

Abstract

Let O_K be the ring of integers of a finite extension K of Q_p. Given two reflexive F-gauges on O_K, we show that for large enough n, the mod p^n-reductions of their first syntomic cohomology groups, which might be regarded as a refinement of local Bloch--Kato Selmer groups, are isomorphic if and only if the mod p^{2n}-reductions of their attached Breuil--Kisin modules with G_K-actions and Nygaard filtrations are isomorphic.