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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 $\mathbb C(A_2(N))$?

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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 $g(C)=0$.

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  • $\begingroup$ A small remark: Maarten only asked for a rational map from $A_g(n)$ to $C$, but the same argument and a tiny bit of mixed Hodge theory rules this out: $H^1(C)$ injects into $W_1H^1(U)$ for some Zariski open $U \subset A_g(n)$, and $W_1H^1(A_g(n)) \to W_1H^1(U)$ is onto by a result in Hodge II. $\endgroup$ – Dan Petersen Jan 8 '14 at 15:02
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    $\begingroup$ You don't need mixed Hodge theory, any rational map to a curve of genus $\geq 1$ is a morphism -- that's a relatively elementary result. $\endgroup$ – abx Jan 8 '14 at 15:26

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