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MULTIPLICITY ONE AT FULL CONGRUENCE LEVEL
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 21 / Issue 2 / March 2022
- Published online by Cambridge University Press:
- 15 May 2020, pp. 637-658
- Print publication:
- March 2022
-
- Article
- Export citation
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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].
