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Let A_2 be the moduli stack of principally polarized abelian surfaces and V asmooth l-adic sheaf on A_2 associated to an irreducible rational finitedimensional representation of Sp(4). We give an explicit expression for thecohomology of V in any degree in terms of Tate type classes and Galoisrepresentations attached to elliptic and Siegel cusp forms. This confirms aconjecture of Faber and van der Geer. As an application we prove a dimensionformula for vector-valued Siegel cusp forms for Sp(4,Z) of weight three, whichhad been conjectured by Ibukiyama.

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