A theta relation in genus 4

Abstract

The $2^{g}$ theta constants of second kind of genus $g$ generate a graded ring of dimension $g(g+1)/2$. In the case $g \geq 3$ there must exist algebraic relations. In genus $g = 3$ it is known that there is one defining relation. In this paper we give a relation in the case $g = 4$. It is of degree $24$ and has the remarkable property that it is invariant under the full Siegel modular group and whose $\Phi$-image is not zero. Our relation is obtained as a linear combination of code polynomials of the $9$ self-dual doubly-even codes of length $24$.