[Paper Review] Multiplicity one for the mod $p$ cohomology of Shimura curves: the tame case

Yongquan Hu, Haoran Wang|arXiv (Cornell University)|Aug 29, 2016

Algebraic Geometry and Number Theory7 references5 citations

TL;DR

This paper establishes a multiplicity one property for the mod $p$ cohomology of Shimura curves in the tame case: when the local Galois representation $\overline{\rho}$ at a place $v$ above $p$ is tamely ramified and generic, the $K_1$-invariant subspace of the associated $\mathrm{GL}_2(F_v)$-representation on the mod $p$ cohomology is isomorphic to the representation $D_0(\overline{\rho})$ constructed by Breuil and Paškūnas, proving that this subspace depends only on $\overline{\rho}$ and satisfies multiplicity one.

ABSTRACT

Let $F$ be a totally real field, $\mathfrak{p}$ an unramified place of $F$ dividing $p$ and $\overline{r}: \mathrm{Gal}(\overline{F}/F) ightarrow\mathrm{GL}_2(\overline{\mathbb{F}}_p)$ a continuous irreducible modular representation. The work of Buzzard, Diamond and Jarvis associates to $\overline{r}$ an admissible smooth representation of $\mathrm{GL}_2(F_\mathfrak{p})$ on the mod $p$ cohomology of Shimura curves attached to indefinite division algebras which split at $\mathfrak{p}$. When $\overline{r}|_{\mathrm{Gal}(\overline{F_\mathfrak{p}}/F_\mathfrak{p})}$ is tamely ramified and generic (and under some technical assumptions), we determine the subspace of invariants of this representation under the principal congruence subgroup of level $\mathfrak{p}$. In particular, it depends only on $\overline{r}|_{\mathrm{Gal}(\overline{F_\mathfrak{p}}/F_\mathfrak{p})}$ and verifies a multiplicity one property.

Motivation & Objective

• To determine the structure of the $K_1$-invariant subspace of the $\mathrm{GL}_2(F_v)$-representation on the mod $p$ cohomology of Shimura curves.
• To establish a multiplicity one property for this subspace under tamely ramified and generic local Galois representations.
• To verify that the $K_1$-invariant subspace depends only on the local Galois representation $\overline{\rho}$ at $v$, not on global data.
• To provide a local constraint on the hypothetical mod $p$ local Langlands correspondence via cohomological realization.

Proposed method

• Uses the minimal fixed determinant patching functor $M^\min_{\overline{r},\infty}$ to analyze the $\mathrm{GL}_2(F_v)$-representation $\pi^D_v(\overline{r})$ on mod $p$ cohomology.
• Applies the local criterion from Corollary 2.29 to bound the dimension of $\mathrm{Hom}$-spaces between $\Gamma = \mathrm{GL}_2(k_v)$-representations and $\pi^D_v(\overline{r})^{K_1}$.
• Relies on the construction of $D_0(\overline{\rho})$ via Breuil-Paškūnas for tame inertial types and their associated lattices.
• Uses the weight part of Serre's conjecture and the structure of Jordan-Hölder constituents of $\overline{V(\tau_v)}$ to analyze the socle of $\pi^D_v(\overline{r})^{K_1}$.
• Applies results from [13] on pro-$p$ Iwahori fixed vectors to reduce the problem to the minimal patching framework.
• Employs the theory of tame types and their lattices to relate $\mathrm{Hom}$-spaces to the structure of coherent sheaves over deformation rings.

Experimental results

Research questions

• RQ1 Does the $K_1$-invariant subspace of the $\mathrm{GL}_2(F_v)$-representation on mod $p$ cohomology of Shimura curves satisfy a multiplicity one property when $\overline{\rho}$ is tamely ramified and generic?
• RQ2 Can the $K_1$-invariant subspace be described purely in terms of the local Galois representation $\overline{\rho}$ at $v$?
• RQ3 Is the $K_1$-invariant subspace isomorphic to the representation $D_0(\overline{\rho})$ constructed by Breuil and Paškūnas?
• RQ4 Does the subspace $\pi^D_v(\overline{r})^{K_1}$ embed into the direct sum of injective envelopes of its Jordan-Hölder constituents?
• RQ5 How does the minimal patching functor $M^\min_{\overline{r},\infty}$ constrain the structure of $\pi^D_v(\overline{r})^{K_1}$?

Key findings

• The $K_1$-invariant subspace $\pi^D_v(\overline{r})^{K_1}$ is isomorphic to $D_0(\overline{\rho})$ as a $\mathrm{GL}_2(\mathcal{O}_{F_v})$-representation when $\overline{\rho}$ is tamely ramified and generic.
• The dimension of $\mathrm{Hom}_{\Gamma}(V_{\sigma}^\circ / \varpi_E V_{\sigma}^\circ, \pi^D_v(\overline{r})^{K_1})$ is exactly one for each $\sigma$ in the set $\mathscr{D}(\overline{\rho}_v)$, confirming multiplicity one.
• The subspace $\pi^D_v(\overline{r})^{K_1}$ embeds into the direct sum of injective envelopes of its Jordan-Hölder constituents, as required by the local criterion.
• The coherent sheaf $M^\min_{\overline{r},\infty}((V_{\sigma})^\circ)$ is free of rank one over $R^\min_{\infty,\tau_v}$, implying the associated $\mathrm{Hom}$-space has dimension at most one.
• The $\mathrm{GL}_2(\mathcal{O}_{F_v})$-socle of $\pi^D_v(\overline{r})^{K_1}$ is isomorphic to $\bigoplus_{\sigma \in \mathscr{D}(\overline{\rho}_v)} \sigma$, matching the prediction of the Buzzard-Diamond-Jarvis conjecture.
• The result holds under mild Taylor-Wiles type hypotheses and extends to the definite quaternion algebra case, confirming consistency across settings.