Multiplicity one for wildly ramified representations

Abstract

Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}:G_F\to {\mathrm{GL}}_2({\overline{\mathbb{F}}}_p)$ 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 Paškūnas. In particular, it depends only on the local representation $\overline{r}{|}_{G_{F_v}}$, and its Jordan-Hölder factors appear with multiplicity one. This builds on and extends work of the author with Morra and Schraen and independently 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.

20 pages

Cited by in corpus (4)

• Diagrams in the mod $p$ cohomology of Shimura curves
• Conjectures and results on modular representations of ${\mathrm{GL}}_n(K)$ for a $p$-adic field $K$
• Serre weight conjectures for $p$-adic unitary groups of rank 2
• On the constituents of the mod $p$ cohomology of Shimura curves