# Albanese of Siegel modular variety $\mathcal{A}_2$

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?

• 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.
– Bear
Oct 24 '16 at 18:23
• 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.
– znt
Oct 24 '16 at 22:22
• 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. 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.
• 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 $N>>1$. Jan 21 '19 at 16:14