Torsion graded pieces of Nyggard filtration for crystalline representation

TL;DR

This paper investigates the structure of torsion in the graded pieces of the Nyggard filtration for crystalline Galois representations, revealing specific conditions under which nontrivial p-torsion occurs.

Contribution

It provides a detailed analysis of the torsion structure in the Nyggard filtration for crystalline representations, connecting torsion to Hodge-Tate weights and filtration indices.

Findings

Nontrivial p-torsion occurs only at specific graded pieces related to Hodge-Tate weights.

The i-graded piece has nontrivial p-torsion only if i = r_j + m p for some weight r_j and integer m.

The result clarifies the torsion behavior in the integral filtration of crystalline Galois representations.

Abstract

Let $K$ be a unramified $p$-adic field with the absolute Galois group $G_K$ and $T$ a crystalline $\mathbb Z_p$-representation of $G_K$. We study the graded pieces of integral filtration on $D_{\rm dR}(T)$ given by Nyggard filtration of the attached Breuil-Kisin module of $T$. We show that the $i$-graded piece has nontrivial $p$-torsion only if $ i = r_j +m p$ for a Hodge-Tate weight $ r_j$ of $T$ and $m$ a positive integer.