Cohomology of local systems on the moduli of principally polarized abelian surfaces

Dan Petersen

Vol. 275 (2015), No. 1, 39–61

DOI: 10.2140/pjm.2015.275.39

Abstract

Let ${\mathcal{A}}_2$ be the moduli stack of principally polarized abelian surfaces. Let $\mathbb{V}$ be a smooth $\ell$-adic sheaf on ${\mathcal{A}}_2$ associated to an irreducible rational finite-dimensional representation of $Sp\left(4\right)$. We give an explicit expression for the cohomology of $\mathbb{V}$ in any degree in terms of Tate-type classes and Galois representations attached to elliptic and Siegel cusp forms. This confirms a conjecture of Faber and van der Geer. As an application we prove a dimension formula for vector-valued Siegel cusp forms for $Sp\left(4,\mathbb{Z}\right)$ of weight three, which had been conjectured by Ibukiyama.

Keywords

abelian surfaces, zeta functions, Galois representations, cohomology of Shimura varieties

Mathematical Subject Classification 2010

Primary: 11F46, 11F67, 11F75, 11G18, 14K10

Milestones

Received: 20 December 2013

Revised: 16 October 2014

Accepted: 29 October 2014

Published: 12 April 2015