A theorem of Igusa on a ring of modular forms of genus two and half-integral weight (Q1913324)

The well-known theorem of Igusa on the structure of the graded ring of Siegel modular forms \({\mathcal A}_n\) is reproved in a generalized version for genus two without using algebraic-geometric methods. Guided by the elliptic case due to Busam et al., the author thoroughly studies the decomposition of \({\mathcal A}_2\) into eigenspaces induced by the characters of the finite group \(\Gamma_2 [2]/ \pm \Gamma_2 [4, 8]\) and the zero-locus of the product \(\Pi \vartheta_m\) of the theta-nullwerte over all even characteristics \(m\), which turns out to be the union of the \(\Gamma_2\)-translates of the diagonal of the Siegel upper half space \(S_2\). Finally an induction based on the weight-decreasing mapping \(f\mapsto f/ \vartheta_m\) yields the desired result, namely, that any modular form for the Igusa modular group \(\Gamma_2 [4, 8]\) is a polynomial in the variables \(\vartheta_m\).