The Burkhardt Group And Modular Forms 2

AbstractIn the first paper we determined the graded algebra A(Γ2[3]) of Siegel modular forms of genus 2 and level 3. The ring A(Γ2[3]) turned out to be the normalization of the algebra C[A1, . . . , A5] where the dening relation is a certain polynomial of degree 18. This polynomial turned out to be the dual of the Burkhardt quartic polynomial. In other approaches to the Siegel modular variety of genus 2 and level 3, the Burkhardt quartic itself comes into the play. From our point of view this picture must be visible if we consider a subgroup Γ’2[3] ⊂2[3] of index 2. In this paper we will see that the graded algebra A(Γ’2[3]) can be determined. We will determine generators of this algebra, generators of the ideal of relations between them and the Hilbert function. We will describe some forms as additive lifts of elliptic modular forms in the sense of Borcherds. These constructions have been done in an unpublished paper of Klaus Merz.