The low degree cohomology of compactifications of $A_g$

TL;DR

The paper determines the low-degree $\ell$-adic intersection cohomology of symplectic local systems on the Satake compactification $A_g^{\mathrm{Sat}}$, and shows that only a finite set of irreducible Galois representations occur in the cohomology of any nonsingular toroidal compactification of $A_g$ or of $X_{g,s}$. It expresses these contributions via Arthur–Langlands parameters and their spin lifts, with the Satake data controlling all strata contributions, and proves independence from the toroidal choice. It then provides sharp Tate vs non-Tate dichotomies, computes holomorphic form spaces, and derives Hodge-theoretic consequences, using a stratified-weight spectral sequence and Leray analysis of boundary strata. The results weave automorphic, Galois, and geometric structures into a coherent framework for understanding how Langlands theory constrains the cohomology of moduli spaces of abelian varieties, with explicit classifications in many low-weight cases and actionable corollaries for holomorphic forms and interior cohomology.

Abstract

We compute the low degree $\ell$-adic intersection cohomology of symplectic local systems on the Satake compactification of the moduli space $A_g$ of principally polarized abelian varieties. We prove that only a small finite list of irreducible Galois representations can appear in the low degree cohomology of any nonsingular toroidal compactification of $A_g$ or $X_{g,s}$, the $s$-fold fiber product of the universal abelian variety. We give several applications, including to spaces of holomorphic forms on toroidal compactifications and to the cohomology of the interior. In particular, we give a complete characterization of when the cohomology of $X_{g,s}$, or one of its toroidal compactifications, is of Tate type. The result is independent of the choice of toroidal compactification.

Theorems & Definitions (51)

• Theorem 1.1: Chenevier--Lannes
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Theorem \ref{['thm IH tables']}
• Remark 1.5
• Remark 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 1.9
• Theorem 1.10