The topology of $A_g$ and its compactifications

TL;DR

This paper surveys the cohomology of the moduli space of principally polarized abelian varieties and its compactifications, focusing on small genus cases, stabilization phenomena, and computational methods involving automorphic representations.

Contribution

It provides a comprehensive overview of both geometric and representation theoretic approaches, including new computational techniques based on trace formulae and automorphic classification.

Findings

Cohomology computations for small genus cases.

Stabilization results in the cohomology of $A_g$.

Development of computational methods using trace formulae.

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.