Galois Lattices and Strongly Divisible Lattices in the Unipotent Case

TL;DR

This paper establishes a fundamental anti-equivalence between unipotent strongly divisible lattices and Galois stable lattices in unipotent semi-stable representations, completing a key part of Breuil's conjecture in p-adic Hodge theory.

Contribution

It proves the last remaining part of Breuil's Conjecture by demonstrating an anti-equivalence in the unipotent case of p-adic Hodge structures.

Findings

Proves anti-equivalence between categories of lattices

Completes the proof of Breuil's Conjecture in the unipotent case

Advances understanding of p-adic Hodge theory structures

Abstract

Let p be a prime. We prove that there is an anti-equivalence between the category of unipotent strongly divisible lattices of weight p-1 and the category of Galois stable Z_p lattices in unipotent semi-stable representations with Hodge-Tate weights in {0, ..., p-1}. This completes the last remaining piece of Breuil's Conjecture(Conjecture 2.2.6 in [Bre02]).