Aleksander Shmakov: Cohomology of local systems on Siegel threefolds

Abstract:

The cohomology of Shimura varieties is a powerful tool for establishing cases of the Langlands correspondence, for example by locating Galois representations attached to modular forms in the cohomology of local systems on modular curves. Beyond modular curves, the story is more complicated: by work of Harder and Petersen one can explicitly compute the cohomology of local systems on the moduli of principally polarized Abelian surfaces, but explicit computations for other Siegel threefolds have remained elusive. Work of Bergström-Faber-van der Geer provides precise conjectures about the cohomology of local systems on the moduli of principally polarized Abelian surfaces with full level 2 structure, but little is known beyond this. In this talk I will summarize some aspects of my ongoing work which resolves these conjectures of Bergström-Faber-van der Geer and provides explicit computations of the cohomology of local systems on Siegel threefolds with square-free parahoric level structure in general, including what remains to be done, and possible future directions.