Breuil's Lattice Conjecture for GL2(K)

TL;DR

This paper proves Breuil's lattice conjecture for GL2(K) with higher Hodge-Tate weights, showing the lattice's dependence on Galois representations and establishing cyclicity of patched modules under certain conditions.

Contribution

It extends Breuil's lattice conjecture to higher Hodge-Tate weights for GL2(K) and introduces new structure theorems and explicit deformation ring computations.

Findings

Lattice inside locally algebraic types depends only on Galois representation.

Patched modules with irreducible cosocle are cyclic.

Structure theorem for mod p representations of GL2(O_K).

Abstract

We prove Breuil's lattice conjecture for higher Hodge-Tate weights in the case of $\mathrm{GL}_2(K)$ where $K$ is an unramified extension of $\mathbb{Q}_p$. More precisely, under some genericity conditions, we show that the lattice inside a locally algebraic type induced by the completed cohomology of a $U(2)$-arithmetic manifold depends only on the Galois representation at places above $p$ for arbitrary Hodge-Tate weights, which are small relative to $p$. We further prove that the patched modules of all lattices inside the locally algebraic types with irreducible cosocle are cyclic. One key input of the paper is a structure theorem for mod $p$ representations of $\mathrm{GL}_2(\mathcal{O}_K)$, which are residually multiplicity free and of finite length. Another input is an explicit computation of universal framed Galois deformation rings, which parameterize potentially crystalline lifts with fixed tame inertial types and higher Hodge-Tate weights.