Integral structures on the finite part $H^1_f(K, V)$ of a crystalline representation

TL;DR

This paper investigates integral structures within the finite part of crystalline Galois representations over unramified extensions, utilizing a specialized auxiliary ring to connect p-adic Hodge theory, Fontaine-Laffaille modules, and local L-functions.

Contribution

It introduces an auxiliary ring $A_{exp}$ to analyze integral structures and establishes new exact sequences linking p-adic Hodge theory, Fontaine-Laffaille modules, and Bloch-Kato sequences.

Findings

Constructed an exact sequence involving $A_{exp}$ and Frobenius actions.

Linked integral structures to Fontaine-Laffaille modules and Bloch-Kato sequences.

Computed the integral finite part of crystalline representation lattices and related it to local L-functions.

Abstract

We study integral structures of crystalline representations over an unramified extension $K / \mathbb{Q}_p$ with the help of an auxillary ring $A_{\textrm{exp}}$. This ring has the nice property that it contains the the fundamental period (and its inverse) of $p$-adic Hodge theory, up to powers of $p$. We establish an exact sequence using $A_{\textrm{exp}}$ and Frobenii on its filtration, give a link to Fontaine-Laffaille modules and the Bloch-Kato fundamental exact sequence and finally compute the integral finite part of a lattice of a crystalline representation, giving a connection to the local $L$-function of $V$.