MULTIPLICITY ONE AT FULL CONGRUENCE LEVEL. (15th March 2022) - Le, Daniel; Morra, Stefano; Schraen, Benjamin
Abstract: Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}:G_{F}→ \text{GL}_{2}(\overline{F}_{p})$ be a modular Galois representation that satisfies the Taylor–Wiles hypotheses and is tamely ramified and generic at a place $v$ above $p$ . Let $\mathfrak{m}$ be the corresponding Hecke eigensystem. We describe the $\mathfrak{m}$ -torsion in the $\text{mod}\, p$ cohomology of Shimura curves with full congruence level at $v$ as a $\text{GL}_{2}(k_{v})$ -representation. In particular, it only depends on $\overline{r}|_{I_{F_{v}}}$ and its Jordan–Hölder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic $\text{GL}_{2}(F_{q})$ -projective envelopes and the multiplicity one results of Emerton, Gee and Savitt [Lattices in the cohomology of Shimura curves, Invent. Math. 200 (1) (2015), 1–96].