In
mathematics, the Schottky form or Schottky's invariant is a
Siegel cusp form ''J'' of degree 4 and weight 8, introduced by 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
In mathematics, the Siegel upper half-space of degree ''g'' (or genus ''g'') (also called the Siegel upper half-plane) is the set of ''g'' × ''g'' symmetric matrices over the complex numbers whose imaginary part is positive definite. It ...
corresponding to 4-dimensional
abelian varieties
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group law th ...
that are the Jacobian varieties of genus 4 curves). showed that it is a multiple of the difference θ
4(''E''
8 ⊕ ''E''
8) − θ
4(''E''
16) of the two genus 4 theta functions of the two 16-dimensional even unimodular lattices and that its divisor of zeros is irreducible. 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
In mathematics, the Ikeda lift is a lifting of modular forms to Siegel modular forms. The existence of the lifting was conjectured by W. Duke and Ö. Imamoḡlu and also by T. Ibukiyama, and the lifting was constructed by . It generalized the S ...
.
References
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*{{citation, last=Schottky, first= F.
, title=Über die Moduln der Thetafunktionen, jfm= 34.0506.03
, journal=Acta Math., volume= 27, pages= 235–288 , year=1903, doi=10.1007/bf02421309, doi-access=free
Automorphic forms