A remark on a theorem of Runge (Q1127971)

In two remarkable papers \textit{B. Runge} [J. Reine Angew. Math. 436, 57-85 (1993; Zbl 0772.11015), Nagoya Math. J. 138, 179-197 (1995; Zbl 0824.11031)] determined the graded ring of Siegel modular forms of genus 3 for the congruence subgroup \(\Gamma_3(2,4)\). To this end he showed that the closure of the natural image of \(\Gamma_3(2,4)\setminus \mathbb{S}_3\) in \(\mathbb{P}^7\) by means of 8 theta constants is normal. In the paper under review the authors give a different proof of the latter fact by using the Macauly computer algebra program. This also yields a new proof of Runge's result. As an application they show that each cusp form of weight \(\leq 4\) for \(\Gamma_3(2,4)\) vanishes.