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) 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 
• Igusa, Jun-ichi (1982), "On the irreducibility of Schottky's divisor", J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (3): 531–545, http://jairo.nii.ac.jp/0021/00004593/en 
• 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 
• Schottky, F. (1888), "Zur Theorie der Abel'schen Functionen von vier Variabeln", Journal für die Reine und Angewandte Mathematik 102: 304–352, http://resolver.sub.uni-goettingen.de/purl?GDZPPN002160641 
• Schottky, F. (1903), "Über die Moduln der Thetafunktionen", Acta Math. 27: 235–288, doi:10.1007/bf02421309