Multiplicity one for wildly ramified representations

TL;DR

This paper proves a multiplicity one result for certain mod p cohomology representations of Shimura curves, showing they depend only on local Galois representations and extend previous tame ramification results.

Contribution

It establishes multiplicity one for wildly ramified cases, using new intersection theory methods on deformation rings, extending prior tame ramification results.

Findings

$rak{m}$-torsion matches Breuil-Paskunas representation $D_0(ar{r}|_{G_{F_v}})$

Dependence solely on local Galois representation $ar{r}|_{G_{F_v}}$

Jordan-Hölder factors appear with multiplicity one

Abstract

Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}: G_F \rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a modular Galois representation which satisfies the Taylor-Wiles hypotheses and is generic at a place $v$ above $p$. Let $\mathfrak{m}$ be the corresponding Hecke eigensystem. Then the $\mathfrak{m}$-torsion in the mod $p$ cohomology of Shimura curves with full congruence level at $v$ coincides with the $\mathrm{GL}_2(k_v)$-representation $D_0(\overline{r}|_{G_{F_v}})$ constructed by Breuil and Pa\v{s}k\={u}nas. In particular, it depends only on the local representation $\overline{r}|_{G_{F_v}}$, and its Jordan-H\"older factors appear with multiplicity one. This builds on and extends work of the author with Morra and Schraen and independently of Hu-Wang, which proved these results when $\overline{r}|_{G_{F_v}}$ was additionally assumed to be tamely ramified. The main new tool is a method for computing Taylor-Wiles patched modules of integral projective envelopes using multitype tamely potentially Barsotti-Tate deformation rings and their intersection theory.