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$\mathcal{A}_g(N)$ is the moduli space of principally polarized abelian varieties with a level $N$ structure.

Set $C_g=\displaystyle{\lim_{\rightarrow}} H^3(\mathcal{A}_g(N), \mathbb{F}_p)$ where the limit is taken over $N$.

Is $C_g$ a finite $\mathbb{F}_p$-module for sufficiently large $g$?

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  • $\begingroup$ What kind of cohomology are you taking here? Topological (Betti) cohomology of $\mathcal{A}_g(N)(\mathbb{C})$, or something fancier? $\endgroup$
    – David Loeffler
    Commented Oct 13, 2020 at 13:38
  • $\begingroup$ singular cohomology of complex points $\endgroup$
    – user163668
    Commented Oct 13, 2020 at 13:41

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