Multiplicity One at Full Congruence Level

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].

Multiplicity one for wildly ramified representations

“…When ρ is tamely ramified, [HW18,LMS16] show that the patched modules of projective envelopes of irreducible F[GL 2 (k v )]-modules are cyclic modules by describing the submodule structure of these projective envelopes and using the Nakayama method of [EGS15] (cf. Proposition 4.6).…”

Conjectures and results on modular representations of $\mathrm{GL}_n(K)$ for a $p$-adic field $K$

“…Remark 3.4.2.3. In the proof of Theorem 3.4.2.1, and hence also in Corollary 3.4.2.2, one only needs the slightly weaker bounds 1 ≤ r i ≤ p − 4 (and 2 ≤ r 0 ≤ p − 3 if r ṽ is irreducible) in the genericity conditions (viii) on r ṽ (or equivalently ρ) in §3.4.1 (these bound are used in [LMS,§4] which is used in the proof of [BHH + , Prop.8.2.5]).…”

Diagrams in the mod $p$ cohomology of Shimura curves

“…Extension graph. Recall that ω j is the fundamental weight of Λ W which is nonzero in embedding j, and that for a subset J ⊂ J ρ we write ω J = j∈J ω j (this differs slightly from the notation in [LMS,§2]). Our genericity condition on ρ implies that s ρ ω J ∈ Λ µ W for all J.…”

“…More precisely, [Bre14] shows that D(π glob (ρ)) contains one of the 0-diagrams attached to ρ conditional on a conjecture later established in [EGS15] (under a slightly stronger genericity hypothesis). Building on this, the sequence of works [HW18, LMS,Le19] shows that in fact this inclusion is an isomorphism (subject to still stronger genericity hypotheses). However, since this family of 0-diagrams is infinite when f > 1, one may still ask whether D(π glob (ρ)) depends on global choices.…”