Vector valued Siegel modular forms of symmetric tensor weight of small degrees (Q2839763)

The results here were announced long ago (in 2001 and 2002), but in Japanese and without complete proofs. The purpose of the present paper is to present them in a more usable form. The topic is the collection of modular forms \(A_{k,j}(\Gamma_2)\) of weight \(\det^k\,\text{Sym}(j)\) for \(\Gamma_2=\mathrm{Sp}_4(\mathbb Z)\), where \(\text{Sym}(j)\) is the symmetric tensor representation of degree~\(j\). The case \(j=0\) is the case of usual (scalar-valued) Siegel modular forms. It is convenient to deal with the cases \(k\) odd and \(k\) even separately, looking at \(A_{\text{sym}(j)}^{\text{even}}(\Gamma_2)=\bigoplus_{k\text{even}}A_{k,j}(\Gamma_2)\) and similarly \(A_{\text{sym}(j)}^{\text{odd}}(\Gamma_2)\). For fixed \(j\) these are modules over \(A^{\text{even}}(\Gamma_2)=A_{\text{sym}(0)}^{\text{even}}(\Gamma_2)\).NEWLINENEWLINEThe main theorem gives generators and relations for these modules for small~\(j\). Specifically, \(A_{\text{sym}(2)}^{\text{even}}(\Gamma_2)\) has four generators with \(k=21,\,23,\,27,\,29\) and one relation, and \(A_{\text{sym}(4)}^{\text{even}}(\Gamma_2)\), \(A_{\text{sym}(4)}^{\text{odd}}(\Gamma_2)\) and \(A_{\text{sym}(6)}^{\text{even}}(\Gamma_2)\) are free modules with five, five and seven generators respectively. The generators are given explicitly, as Eisenstein series or via Rankin-Cohen operators: there is a helpful discussion of the latter to the extent that they are needed here.