The Siegel modular forms of genus 2 with the simplest divisor

We prove that there exist exactly eight Siegel modular forms with respect to the congruence subgroups of Hecke type of the paramodular groups of genus two vanishing precisely along the diagonal of the Siegel upper half-plane. This is a solution of a problem formulated during the conference “Black holes, Black Rings and Modular Forms” (ENS, Paris, August 2007). These modular forms generalize the classical Igusa form and the forms constructed by Gritsenko and Nikulin in 1998. Introduction: dd-modular forms The first cusp form for the Siegel modular group Sp2(Z) is the Igusa form Ψ10. In fact Ψ10 = ∆5 where ∆5 is the product of the ten even thetaconstants (see [F]). This modular form has a lot of remarkable properties. One of the main features of ∆5 is that it vanishes (with order one) precisely along the diagonal H1 = {( τ 0 0 ω ) , τ, ω ∈ H1 } ⊂ H2 of the Siegel upper half-plane H2 = {Z = ( τ z z ω ) ∈ M2(C), Im(Z) > 0}. It is known that ∆5 determines a Lorentzian Kac–Moody super Lie algebra of Borcherds type. See [GN1]–[GN2] where two lifting constructions of ∆5 were proposed ∆5(Z) = Lift(η(τ)θ(τ, z)) = B(φ0,1)(Z) where η is the Dedekind eta-function and θ is the Jacobi theta-series (see (7)). This relation gives the two parts of the denominator identity for the Borcherds algebra determined by ∆5. There exists a geometric interpretation of this identity in terms of the arithmetic mirror symmetry (see [GN4]). Moreover 2φ0,1 is the elliptic genus of a K3 surface and Ψ−2 10 is related to the so-called second quantized elliptic genus of K3 surfaces (see [DMVV], [G4]). These facts explain the importance of ∆5 in the theory of strings (see [DVV], [Ka]). During the conference “Black holes, Black Rings and