On a ring of modular forms related to the theta gradients map in genus 2

The level moduli space of ppavs with a level structure Ag4,8 can be mapped to the projective space by means of the gradients of odd theta functions. Although this map is generically injective for g⩾3, it is not injective in the genus 2 case. However, there exists a congruence subgroup Γ, contained in Γ2(2,4) and containing Γ2(4,8), such that the theta gradient map factors on the quotient AΓ of the Siegel upper half-plane by the group Γ and the new map is injective on AΓ; we provide a description of this group together with some of its properties. We also prove a structure theorem for the ring of modular forms A(Γ) with respect to Γ. We finally provide a set of generators for the ideal of cusp forms S(Γ) to give an algebraic description of the desingularization ProjS(Γ) of the Satake compactification ProjA(Γ).