On some Siegel threefold related to the tangent cone of the Fermat quartic surface

Let Z be the quotient of the Siegel modular threefold A^sa(2, 4, 8) which has been studied by van Geemen and Nygaard. They gave an implication that some 6-tuple F_Z of theta constants which is in turn known to be a Klingen type Eisenstein series of weight 3 should be related to a holomorphic differential (2, 0)-form on Z. The variety Z is birationally equivalent to the tangent cone of Fermat quartic surface in the title. In this paper we first compute the L-function of two smooth resolutions of Z. One of these, denoted by W, is a kind of Igusa compactification such that the boundary ∂W is a strictly normal crossing divisor. The main part of the L-function is described by some elliptic newform g of weight 3. Then we construct an automorphic representation π of GSp₂(A) related to g and an explicit vector E_Z sits inside π which creates a vector valued (non-cuspidal) Siegel modular form of weight (3, 1) so that F_Z coincides with E_Z in H^2,0(∂W) under the Poincaré residue map and various identifications of cohomologies.