MULTIPLICITY ONE AT FULL CONGRUENCE LEVEL. (15th March 2022)

• Record Type: Journal Article
• Authors: Le, Daniel; Morra, Stefano; Schraen, Benjamin
• Is Part Of: Journal of the Institute of Mathematics of Jussieu. Volume 21:Number 2(2022)
• Page Start: 637
• Page End: 658
• Publication Date: 2022-03-15
• Subjects: multiplicity one -- cohomology of Shimura curves -- mod p Langlands program -- modular representations of GL2(Fq); 11F33 -- 11F80 -- 20C33; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S1474748020000225
• Languages: English
• ISSNs: 1474-7480

Abstract

Abstract: Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbf{F}}_{p})$ 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 $\text{GL}_{2}(\mathbf{F}_{q})$ -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].