Schottky’s Invariant and Quadratic Forms
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 4 2 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 4 2 , f 8 and also by f 4 2 , 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.
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- Igusa, J.: Geometric and analytic methods in the theory of theta functions. Proc. Bombay Colloq. on Algebraic Geometry (1968), 241–253.Google Scholar
- Igusa, J.: Theta Functions. Grund. math. Wiss. 194, Springer-Verlag, 1972.Google Scholar
- Krazer, A.: Lehrbuch der Thetafunktionen. B.G. Teubner, Leipzig, 1903.Google Scholar
- Schottky, F.: Zur Theorie der Abelschen Functionen von vier Variabeln, Crelles J. 102 (1888), 304–352.Google Scholar
- Siegel, C.L.: Über die analytische Theorie der quadratischen Formen, Ann. of Math. 36 (1935), 527–606; Gesam Abhand., Springer-Verlag, 1966, Bd. I, 326–405.Google Scholar