#### Vol. 13, No. 8, 2019

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Multiplicity one for wildly ramified representations

### Daniel Le

Vol. 13 (2019), No. 8, 1807–1827
DOI: 10.2140/ant.2019.13.1807
##### Abstract

Let $F$ be a totally real field in which $p$ is unramified. Let $\stackrel{̄}{r}:{G}_{F}\to {GL}_{2}\left({\overline{\mathbb{F}}}_{p}\right)$ 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 $\phantom{\rule{0.2em}{0ex}}mod\phantom{\rule{0.2em}{0ex}}p$ cohomology of Shimura curves with full congruence level at $v$ coincides with the ${GL}_{2}\left({k}_{v}\right)$-representation ${D}_{0}\left(\stackrel{̄}{r}{|}_{{G}_{{F}_{v}}}\right)$ constructed by Breuil and Paškūnas. In particular, it depends only on the local representation $\stackrel{̄}{r}{|}_{{G}_{{F}_{v}}}$, 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 $\stackrel{̄}{r}{|}_{{G}_{{F}_{v}}}$ 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.

##### Keywords
Galois deformations, mod p Langlands program
Primary: 11S37