Siegel modular forms generated by invariants of cubic hypersurfaces

zbMath0845.11019arXivalg-geom/9403005MaRDI QIDQ4838950

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Publication date: 6 May 1996

Abstract: We give a geometric derivation of Schottky's equation in genus four for the period matrices of Riemann surfaces among all period matrices. The equation arises naturally from the singularity theory of the Gauss map on the theta divisor, and thus generalizes for any genus $gge4$ to a certain ideal of Siegel modular forms vanishing on period matrices of Riemann surfaces. This ideal is generated by modular forms associated to the invariants of cubic forms in $g-1$ variables which vanish on the Fermat cubic.

Full work available at URL: https://arxiv.org/abs/alg-geom/9403005

zbMATH Keywords

theta functionSiegel modular formsSiegel modular groupcubic hypersurfacescubic formsFermat cubicprincipally polarized Abelian varietySchottky modular formGauss map on the theta divisorJacobian of a generic curvezero set of the Fermat ideal

Mathematics Subject Classification ID

Abelian varieties of dimension (> 1) (11G10) Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms (11F46) Modular and Shimura varieties (14G35) Theta functions and curves; Schottky problem (14H42)

Related Items (4)

On Siegel modular forms. Part I. ⋮ A Siegel-Weil formula for automorphic characters: Cubic variation of a theme of Snitz ⋮ Towards the arithmetic degree of line bundles on abelian varieties ⋮ Conic Lagrangian singularities

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