[Paper Review] Compactly supported cohomology and nearby cycle cohomology of open Shimura varieties of PEL type

Naoki Imai, Yoichi Mieda|arXiv (Cornell University)|Sep 22, 2011

Advanced Algebra and Geometry32 references3 citations

TL;DR

This paper establishes that the $G(\mathbb{Q}_p)$-cuspidal parts of compactly supported $\ell$-adic cohomology and nearby cycle cohomology agree for open Shimura varieties of PEL type, even when the integral model has bad reduction at $p$. The key result shows that any discrepancy between these cohomology groups is non-cuspidal at $p$, enabling the use of nearby cycle cohomology to study Galois representations attached to supercuspidal automorphic forms.

ABSTRACT

In this paper, we compare two cohomology groups associated to Shimura varieties of PEL type, which are not proper over the base. One is the compactly supported l-adic cohomology, and the other is the nearby cycle cohomology, namely, the compactly supported cohomology of the nearby cycle complex for the canonical integral model of the Shimura variety over Z_p. We prove that the G(Q_p)-cuspidal part of these cohomology groups are the same, where G denotes the reductive algebraic group naturally attached to the PEL datum. Some applications to unitary Shimura varieties are also given.

Motivation & Objective

• To compare compactly supported $\ell$-adic cohomology and nearby cycle cohomology for open Shimura varieties of PEL type.
• To understand the Galois action on cohomology when the integral model has bad reduction at $p$.
• To establish that the $G(\mathbb{Q}_p)$-cuspidal part of the cohomology is preserved under the nearby cycle construction.
• To extend the use of nearby cycle cohomology as a tool for studying the local Langlands correspondence in non-compact cases.
• To provide applications to unitary Shimura varieties and torsion coefficient systems.

Proposed method

• Use the nearby cycle functor to relate the cohomology of the generic fiber to that of the special fiber of the integral model over $\mathbb{Z}_p$.
• Compare the $G(\mathbb{Q}_p)$-isotypic components of compactly supported cohomology and nearby cycle cohomology via the natural map between them.
• Apply the Hochschild-Serre spectral sequence to analyze the cohomology of the special fiber in terms of automorphic representations.
• Use Mantovan’s formula and the theory of local models to relate cohomology to representations of $J(\mathbb{Q}_p) = D^\times \times \mathbb{Q}_p^\times$.
• Analyze the supercuspidal part of cohomology using the vanishing of supercuspidal components outside degree $n-1$ in the cohomology of local models.
• Extend results to mod-$\ell$ coefficients using techniques from Dat and Shin, particularly the projectivity and vanishing of supercuspidal parts in non-middle degree.

Experimental results

Research questions

• RQ1 Does the nearby cycle cohomology capture the same $G(\mathbb{Q}_p)$-cuspidal representations as the compactly supported cohomology of the generic fiber?
• RQ2 How does the Galois action on cohomology behave when the Shimura variety has bad reduction at $p$?
• RQ3 Can the nearby cycle cohomology be used to study the local Langlands correspondence for non-compact Shimura varieties?
• RQ4 What is the role of the special fiber in controlling the $G(\mathbb{Q}_p)$-representation type in cohomology?
• RQ5 Does the supercuspidal part of cohomology vanish outside degree $n-1$ in the torsion coefficient case?

Key findings

• The $G(\mathbb{Q}_p)$-cuspidal part of the compactly supported cohomology and the nearby cycle cohomology are isomorphic.
• The kernel and cokernel of the natural map between the two cohomology groups contain no irreducible supercuspidal representations of $G(\mathbb{Q}_p)$.
• For an irreducible admissible representation $\Pi$ of $G(\mathbb{A}^\infty)$ with $\Pi_p$ supercuspidal, the $\Pi$-isotypic components of both cohomology groups are isomorphic as $\operatorname{Gal}(\overline{E}_v/E_v)$-modules.
• In the torsion coefficient case, $H^i_c(\operatorname{Sh}_{K^p}, \overline{\mathbb{F}}_\ell)[\pi] = 0$ unless $i = n-1$ for any irreducible supercuspidal $\overline{\mathbb{F}}_\ell$-representation $\pi$ of $G(\mathbb{Q}_p)$.
• The supercuspidal part of the cohomology of the special fiber vanishes outside degree $n-1$, due to the vanishing of supercuspidal components in $H^i_c(\mathcal{M}_{\mathrm{LT},\infty}, \overline{\mathbb{F}}_\ell)$ for $i \neq n-1$.
• The nearby cycle cohomology captures the correct Galois representation for supercuspidal automorphic forms, even in the presence of bad reduction.