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
In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to:
*Jacobian matrix and determinant
*Jacobian elliptic functions
*Jacobian variety
*Intermediate Jacobian
In mathematics, the intermediate Jacobian of a compact Kähler m ...
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 projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...
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 form
In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional ''elliptic'' modular forms which are closely related to ...
.
References
*
*
*
*
*{{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