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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].

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
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