The moduli space of (1,11)-polarized abelian surfaces is unirational
Mark Gross and Sorin Popescu
We prove that the moduli space ${\mathcal A}_{11}^{lev}$ of $(1,11)$-polarized abelian surfaces with level structure of canonical type is birational to Klein's cubic hypersurface in $\mathbf{P}^4$. Therefore, ${\mathcal A}_{11}^{lev}$ is unirational but not rational, and there are no $\Gamma_{11}$-cusp forms of weight 3. The same methods also provide an easy proof of the rationality of ${\mathcal A}_{9}^{lev}$.