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Abstract

Abstract. We prove that the moduli space e lev 11 of 1Y 11 -polarized Abelian surfaces with level structure of canonical type is birational to Klein's cubic hypersurface in P 4 . Therefore, e lev 11 is unirational but not rational, and there are no G 11 -cusp forms of weight 3. The same methods also provide an easy proof of the rationality of e lev 9 . Mathematics Subject Classi¢cation (2000). 14K10.Key words. Abelian surfaces, moduli, Kleine's cubic hypersurface.Classical results of Tai, Freitag and Mumford and newer results of O'Grady, Gritsenko, Hulek and Sankaran say that moduli spaces of polarized Abelian varieties are almost always of general type. However, for Abelian varieties of small dimension and polarizations of small degree the situation is different and the corresponding moduli spaces usually have beautiful geometry.In this paper we describe a projective model for the moduli of complex Abelian surfaces with a polarization of type 1Y 11 , with level structure of canonical type. As a direct consequence we obtain the unirationality of this moduli space, which also turns out to be non-rational. However, unirationality already implies that there exist no G 11 -cusp forms of weight 3.Let

Weight one Jacobi forms and umbral moonshine

“…So the cube of X 1,9 defines a holomorphic differential on A 9 . It was shown by O'Grady [33] (see also [34]) that the Satake compactification of A 9 is rational, so there are no cusp forms of weight 3 for Γ 9 . So X 3 1,9 is an example of a paramodular form of an odd weight that is not a cusp form.…”

Weight One Jacobi Forms and Umbral Moonshine

“…It was shown by O'Grady [33] (see also [34]) that the Satake compactification of A 9 is rational, so there are no cusp forms of weight 3 for Γ 9 . So X 3 1,9 is an example of a paramodular form of odd weight that is not a cusp form.…”