Let $A_2(N)$ denote the moduli space of principally polarized abelian surfaces with level $N$ stucture. The absolute Igusa invariants $i_1$ $i_2$ and $i_3$ give three different maps from $A_2(1)$ to $\mathbb P^1$. Now what I want to do is construct a family of rational maps $f_N : A_2(N) \to C(N)$ where all the $C(N)$ are curves, and such that the genus of $C(N)$ eventually goes to $\infty$. Now I don't know wether this is possible at all so my first question is:
Are there $N$ for which there is a (dominant rational) map $A_2(N) \to C$ where $C$ is a curve of nonzero genus?
And my second question is:
Is $\mathbb C(i_k)$ algebraically closed in $\mathbb C(A_2(N))$?