Jacobian varieties of Shimura curves are very interesting objects. For one thing they provide a geometric relation between elliptic curves and modular forms of weight 2 (say we are over $\mathbb{Q}$). So I was wondering what happens when one consider other Shimura varieties, and to start I would be happy to understand the case of $\mathcal{A}_2$.

Are there references that discuss what kind of abelian variety is the albanese of the Siegel moduli space?

Thank you in advance!

  • $\begingroup$ I forgot. As in the case of modular curves, we first want to take some toroidal compactification of the Siegel modular surface and then we consider its albanese. $\endgroup$
    – Bear
    Oct 24 '16 at 18:23
  • 2
    $\begingroup$ I think $A_2$ has dimension 3... The reason Jacobians are so effective for modular curves is that most interesting algebraic automorphic representations contribute to cohomology in degree 1, hence to $H^1$ of the curve, which is $H^1$ of the Jacobian and hence really strongly controlled by the Jacobian. For $Sp(4)$ probably the interesting cohomology is in degree 3 and I'm not so sure how easy it is to construct a natural geometric object which sees $H^3$ of $A_2$ in any "strong" way. $\endgroup$
    – znt
    Oct 24 '16 at 22:22
  • 2
    $\begingroup$ Another way of showing that the albanese of $A_2$ is trivial is by showing that some smooth compactification is rationally connected. This follows essentially from the fact that the Deligne-Mumford compactification of $M_2$ is rationally connected. (The latter follows from the explicit description of $M_2$ as $(\mathbb{A}^1-\{0,1\})^3- \Delta)/Sym_3$.) If this argument seems strange, compare this to the case of $g=1$. In this case, one (and in fact only) smooth compactification of $A_g = A_1$ is given by $\mathbb P^1$ (on the level of coarse spaces), so that its Albanese is trivial. $\endgroup$ Oct 25 '16 at 12:09

Take a look at Sankaran, Fundamental group of locally symmetric varieties, Manuscripta (1995), and references therein. I think that the toroidal compactification of $\mathcal{A}_2$ and related spaces have finite fundamental group, and therefore trivial Albanese.

  • $\begingroup$ The Jacobian of modular curves of higher level are interesting. Perhaps one should consider the Albanese of not $\mathcal{A}_2$ but rather $\mathcal{A}_2[N]$ for some $N>>1$. $\endgroup$ Jan 21 '19 at 16:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.