Schottky’s Invariant and Quadratic Forms

Abstract

In his classical paper on the moduli of 4 dimensional principally polarized abelian varieties Schottky introduced a homogeneous polynomial J of degree 16 in the Thetanullwerte which vanishes at every jacobian point. On the other hand, the analytic class invariants of even quadratic forms in 16 variables with determinant 1 can be written as f ²₄ and f ₈, and they are Siegel modular forms of weight 8 and of an arbitrary degree g. In this paper explicit expressions of J by f ²₄ , f ₈ and also by f ²₄ , the Eisenstein series E ₈ for g=4 are proved. Also an outline of the proof of the fact that f ₄ is the only Siegel modular form of weight 4 and of any given degree g which can be expressed as a polynomial in the Thetanullwerte is given.

This work was partially supported by the National Science Foundation.