Multiplicity one for wildly ramified representations

Abstract

Let $F$ be a totally real field in which $p$ is unramified. Let $\bar{r}:G_F\to {GL}_2\left({\overline{\mathbb{F}}}_p\right)$ 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. We show that the $\mathfrak{m}$-torsion in the $modp$ cohomology of Shimura curves with full congruence level at $v$ coincides with the ${GL}_2\left(k_v\right)$-representation $D_0\left(\bar{r}{|}_{G_{F_v}}\right)$ constructed by Breuil and Paškūnas. In particular, it depends only on the local representation $\bar{r}{|}_{G_{F_v}}$, and its Jordan–Hölder factors appear with multiplicity one. This builds on and extends work of the author with Morra and Schraen and, independently, Hu–Wang, which proved these results when $\bar{r}{|}_{G_{F_v}}$ 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.