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 $\mathbb C(A\_2(N))$?



Maps from the moduli space of abelian surfaces with level stucture to curves
The answer to the first question is negative. It follows from an old result of Matsushima that $b\_1(\mathcal{A}\_g(n))=0$ for $g\geq 2$ (Annals of Math. 75 (1962), 312-330). If there is a dominant map $\mathcal{A}\_g(n)\rightarrow C$, $H^1(C,\mathbb{Q})$ injects into $H^1(\mathcal{A}\_g(n),\mathbb{Q})$, which implies $g(C)=0$.


