The topology of $A_g$ and its compactifications

Abstract

We survey old and new results about the cohomology of the moduli space $A_g$ of principally polarized abelian varieties of genus $g$ and its compactifications. The main emphasis lies on the computation of the cohomology for small genus and on stabilization results. We review both geometric and representation theoretic approaches to the problem. The appendix provides a detailed discussion of computational methods based on trace formulae and automorphic representations, in particular Arthur's endoscopic classification of automorphic representations for symplectic groups.

57 pages. Appendix by Olivier Taïbi. To appear: Proceedings of the Abel Symposium 2017. v3: numbering of theorems changed to agree with the printed version

References in corpus (4)

• Stable cohomology of the perfect cone toroidal compactification of ${\mathcal{A}}_g$
• Formes automorphes et voisins de Kneser des réseaux de Niemeier
• Goresky-Pardon lifts of Chern classes and associated Tate extensions
• The singularities of $A_g^P$