Siegel Modular Forms of Genus 2 and Level 2: Cohomological Computations and Conjectures

In this paper we study the cohomology of certain local systems on moduli spaces of principally polarized abelian surfaces with a level 2 structure that corresponds to prescribing a number of Weierstrass points in case the abelian surface is the Jacobian of a curve of genus 2. These moduli spaces are defined over Z[1/2] and we can calculate the trace of Frobenius on the alternating sum of the étale cohomology groups of these local systems by counting the number of pointed curves of genus 2 with a prescribed number of Weierstrass points that separately or taken together are defined over the given finite field. This cohomology is intimately related to vector-valued Siegel modular forms. Two of the present authors carried out this scheme for local systems on the moduli space A2 of level 1 in [9]. This provided new information on Siegel modular forms and led for example to a precise formulation of a conjecture of Harder about congruences between genus 1 and genus 2 modular forms and also to experimental evidence supporting it, cf. [14, 11]. Here we extend this scheme to level 2 where new phenomena appear. In order to be able to extract information on Siegel modular forms we must subtract the contributions to the cohomology from the boundary, that is, the Eisenstein cohomology, and the endoscopic contributions. We can determine the contribution of the Eisenstein cohomology together with its S6-action for the full level 2 structure and on the basis of our computations we can make precise conjectures on the endoscopic contribution. We also make a prediction about the existence of a vector-valued analogue of the Saito-Kurokawa lift. Assuming these conjectures that are based on ample numerical evidence, we can obtain the traces of the Hecke-operators T (p) for p ≤ 37 on the remaining spaces of ‘genuine’ Siegel modular forms. We present a number of examples of 1-dimensional spaces of eigenforms where these traces coincide with the Hecke eigenvalues to illustrate this. We hope that the experts on lifting and on endoscopy will be able to prove our conjectures.