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Let $\mathcal{A_g}$ be the moduli space of principally polarized abelian varieties over $\mathbb C$, $\mathcal{M_g}$ be the moduli space of smooth projective curves of genus $g$ over $\mathbb C$, and we regard both as Deligne-Mumford stacks or orbifolds. By computations we know their Euler characteristics :

$\chi(\mathcal{A_g})=\chi(\mathcal M_{1})\chi(\mathcal M_{2})...\chi(\mathcal M_{g})=\prod_{k=1}^g \zeta(1-2k)$.

Is there a conceptual explanation of this using Torelli map? Can we update it to an identity of Poincare polynomials?

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    $\begingroup$ These Euler characteristics are not the alternating sums of Betti numbers and therefore aren't related to Poincare polynomials. For instance, they aren't integers. $\endgroup$ – Will Sawin Apr 17 at 2:12
  • $\begingroup$ @WillSawin There are Poincaré polynomials associated with orbifolds. $\endgroup$ – sawdada Apr 17 at 2:53
  • $\begingroup$ @WillSawin More precisely, one can replace $(-1)$ in $\chi(M/G) = \frac1{|G|}\sum_i\sum_{g\in G} (-1)^i\mathrm{Tr}_{H^i(M)}(g^*)$ by $t$ as in mathoverflow.net/questions/51993/… $\endgroup$ – sawdada Apr 17 at 14:25
  • $\begingroup$ But that formula is for the Euler characteristic as an alternating sum of Betti numbers (see step 2 of Johannes Ebert's answer), not the other definition of the Euler characteristic for orbifolds, which gives $\chi(M/G) =\chi(M)/|G|$ and which is used in the zeta function identities you state. $\endgroup$ – Will Sawin Apr 17 at 14:50

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