Parahoric Restriction for GSp(4) and the Inner Cohomology of Siegel Modular Threefolds

Abstract

For irreducible admissible representations of the group of symplectic similitudes GSp(4,F) of genus two over a p-adic number field F, we obtain the parahoric restriction with respect to an arbitrary parahoric subgroup. That means we determine the action of the Levi quotient on the invariants under the pro-unipotent radical in terms of explicit character values. Especially, we get the parahoric restriction of local endoscopic L-packets in terms of lifting data. The inner cohomology of the Siegel modular variety of genus two with an arbitrary l-adic local system admits an endoscopic and a Saito-Kurokawa part under spectral decomposition. For principal congruence subgroups of squarefree level N they define simultaneous representations of the absolute Galois group and the Hecke action of GSp(4;Z/NZ). We decompose them into irreducible constituents and give explicit character values. As an application, we prove the conjectures of Bergstrom, Faber and van der Geer on level two.