Some remarks to a Theorem of van Geemen

TL;DR

This paper provides a new proof and extension of van Geemen's theorem on linear relations among theta nullwerte powers, showing that only the fourth power exhibits such dependencies, with broader implications for theta function relations.

Contribution

It offers a novel proof of van Geemen's theorem and extends the analysis to arbitrary powers of theta functions, identifying the unique case of the fourth power.

Findings

All linear relations between theta nullwerte powers are consequences of quartic Riemann relations.

The only power of theta functions with linear dependencies is the fourth power.

The paper extends the understanding of linear dependencies to arbitrary powers, showing k=4 is unique.

Abstract

In [ Ge], Bert van Geemen computed the dimension of the space of the fourth power of the theta nullwerte. In [SM2], it has been observe that all linear relations between the $\theta_m^4$ are consequences of the quartic Riemann relations. In this note, we want to give a new proof of these result and extend them. In a last section we treat the linear dependencies between arbitrary powers $\vartheta[m]^k$. We will show that $k=4$ is the only case where such dependencies can occur. For this reason, we give a slightly different title: Some remarks to a Theorem of van Geemen