A theta relation in genus 4 (Q2725471)

The \(2^g\) theta constants of second kind generate a ring of dimension \(g(g+1)/2+1\). In the case \(g\leq 2\) they are algebraically independent. In the case \(g=3\) there is a relation in degree 16, which is related to the so-called Schottky relation. In this paper, the authors investigate the case \(g=4\) and find a relation whose degree is 24. The relation has two remarkable properties. Namely, it is invariant under the full Siegel modular group, and it is mapped to a nontrivial relation under Siegel's \(\Phi\)-operator. The proof is based on the theory of codes. The relation is obtained as a linear combination of code polynomials of the 9 self-dual doubly-even codes of length 24.