Schottky form

In mathematics, the Schottky form or Schottky's invariant is a Siegel cusp form J of degree 4 and weight 8, introduced by Friedrich Schottky (1888, 1903) as a degree 16 polynomial in the Thetanullwerte of genus 4. He showed that it vanished at all Jacobian points (the points of the degree 4 Siegel upper half-space corresponding to 4-dimensional abelian varieties that are the Jacobian varieties of genus 4 curves). Igusa (1981) showed that it is a multiple of the difference θ₄(E₈ ⊕ E₈) − θ₄(E₁₆) of the two genus 4 theta functions of the two 16-dimensional even unimodular lattices and that its divisor of zeros is irreducible. Poor & Yuen (1996) showed that it generates the 1-dimensional space of level 1 genus 4 weight 8 Siegel cusp forms. Ikeda showed that the Schottky form is the image of the Dedekind Delta function under the Ikeda lift.

References

Igusa, Jun-ichi (1981), "Schottky's invariant and quadratic forms", E. B. Christoffel (Aachen/Monschau, 1979), Basel-Boston, Mass.: Birkhäuser, pp. 352–362, doi:10.1007/978-3-0348-5452-8_24, ISBN 978-3-7643-1162-9, MR 0661078
Igusa, Jun-ichi (1982) [1981], "On the irreducibility of Schottky's divisor", J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28 (3): 531–545, MR 0656035
Poor, Cris; Yuen, David S. (1996), "Dimensions of spaces of Siegel modular forms of low weight in degree four", Bull. Austral. Math. Soc., 54 (2): 309–315, doi:10.1017/s0004972700017779, MR 1411541
Schottky, F. (1888), "Zur Theorie der Abel'schen Functionen von vier Variabeln", Journal für die Reine und Angewandte Mathematik, 102: 304–352, JFM 20.0488.02
Schottky, F. (1903), "Über die Moduln der Thetafunktionen", Acta Math., 27: 235–288, doi:10.1007/bf02421309, JFM 34.0506.03