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Multiplicity One at Full Congruence Level by Le, Daniel; Morra, Stefano; Schraen, Benjamin is a scholarly article available to read on EtoBox.

What is Multiplicity One at Full Congruence Level about?

Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}: G_F \rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a modular Galois representation which 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 mod $p$ cohomology of Shimura curves with full congruence level at $v$ as a $\mathrm{GL}_2(k_v)$-representation. In particular, it only depends on $\overline{r}|_{I_{F_v}}$ and its Jordan--H\"{o}lder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic $\mathrm{GL}_2(\mathbb{F}_q)$-projective envelopes and the multiplicity one results of \cite{EGS}.

Author

Le, Daniel; Morra, Stefano; Schraen, Benjamin

Published

2016

Language

EN