What is the Brauer group of the moduli space of (p.p.) abelian varieties? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T06:08:05Z http://mathoverflow.net/feeds/question/108211 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108211/what-is-the-brauer-group-of-the-moduli-space-of-p-p-abelian-varieties What is the Brauer group of the moduli space of (p.p.) abelian varieties? Felipe Voloch 2012-09-27T05:23:48Z 2012-09-27T11:30:22Z <p>What is the Brauer group of the moduli space of principally polarized abelian varieties of a given dimension? I am primarily interested in the "open" moduli space, i.e. not a compactification. The question has several levels of generality depending on how general is the ground field (ring even?) but I know nothing, so already the knowing the analytic Brauer group over the complex numbers would be very interesting for me.</p> http://mathoverflow.net/questions/108211/what-is-the-brauer-group-of-the-moduli-space-of-p-p-abelian-varieties/108231#108231 Answer by Jason Starr for What is the Brauer group of the moduli space of (p.p.) abelian varieties? Jason Starr 2012-09-27T11:30:22Z 2012-09-27T11:30:22Z <p>I am posting as an answer instead of a comment, even though I think this might be wrong, because I could not format the weblinks properly in comments.</p> <p>Since the (orbifold) moduli space is a quotient of the Siegel upper half space by the (orbifold) fundamental group $\textbf{Sp}_{2g}(\mathbb{Z})$, it seems to me that the analytic Brauer group of the moduli space should be $\text{Hom}(H_2,\mathbb{Q}/\mathbb{Z})$, where $H_2 = H_2(\textbf{Sp}_{2g}(\mathbb{Z}))$ is the kernel of the universal central extension of <code>$\textbf{Sp}_{2g}(\mathbb{Z})$</code>. According to the proof of Proposition 2.3 of <a href="http://arxiv.org/pdf/1103.1855.pdf" rel="nofollow">Finite quotients of symplectic groups vs mapping class groups</a> by Funar and Pitsch, it is well-known that $H_2$ equals $\mathbb{Z}$ for all $g\geq 3$. So this implies that the analtyic Brauer group is $\mathbb{Q}/\mathbb{Z}$. </p>