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.
Categories: math.NT