van der Geer, Gerard The cohomology of the moduli space of abelian varieties. (English) Zbl 1322.14019 Farkas, Gavril (ed.) et al., Handbook of moduli. Volume I. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-257-2/pbk; 978-1-57146-265-7/set). Advanced Lectures in Mathematics (ALM) 24, 415-457 (2013). From the introduction: In this survey we are concerned with the cohomology of the moduli space of abelian varieties. I have chosen to stick to the moduli spaces of principally polarized abelian varieties, leaving aside the moduli spaces of non-principal polarizations and other variations. The emphasis is on the tautological ring. We discuss the cycle classes of the Ekedahl-Oort stratification, that can be expressed in tautological classes, and discuss differential forms on the moduli space. We also discuss complete subvarieties of \(\mathcal A_g\). Finally, we discuss Siegel modular forms and its relations to the cohomology of these moduli spaces. We sketch the approach developed jointly with Faber and Bergström to calculate the traces of the Hecke operators by counting curves of genus \(\leq 3\) over finite fields, an approach that opens a new window on Siegel modular forms.For the entire collection see [Zbl 1260.14001]. Cited in 4 Documents MSC: 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 14K10 Algebraic moduli of abelian varieties, classification 11G10 Abelian varieties of dimension \(> 1\) 14C15 (Equivariant) Chow groups and rings; motives 14F25 Classical real and complex (co)homology in algebraic geometry Keywords:Siegel modular forms; moduli space; abelian varieties; principally polarized abelian varieties; tautological ring; Ekedahl-Oort stratification PDFBibTeX XMLCite \textit{G. van der Geer}, Adv. Lect. Math. (ALM) 24, 415--457 (2015; Zbl 1322.14019) Full Text: arXiv