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

By Dan Erik Petersen. 2013

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Let A2 be the moduli stack of principally polarized abelian surfaces. Let V be a smooth ‘-adic sheaf on A2 associated to an irreducible rational finitedimensional representation of Sp.4/. We give an explicit expression for the cohomology of 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.4;Z/ of weight three, which had been conjectured by Ibukiyama.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/20f99eeb689f4d4cf17b0603a5936896244de075