Topological Considerations for the Computation of Hecke Operators on the Cohomology of Siegel Modular Threefolds

Abstract

This thesis considers various approaches to the computation of Hecke operators on the cohomology of Siegel modular threefolds, discussing in detail topological phenomena that arise from the explicit reduction theory of Siegel modular threefolds, and transversality issues that occur when attempting to pass to common refinements after Hecke action on a candidate good cover for the Siegel upper half-space of complex dimension three and genus two. We evaluate an initial approach to a computation of Čech cohomology in light of these observations and present a conjecture for the existence of a new, more canonical retract for the Siegel upper half-space. Finally, we discuss first steps towards a possible resolution of this conjecture, considering tilings of the Borel-Serre bordification. Throughout, we draw analogies between the rank two and rank one situations, providing detailed computations and illustrations where possible.