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

Abstract

Let $Z$ be the quotient of the Siegel modular threefold $\mathcal{A}^{{\rm sa}}(2,4,8)$ 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 $\partial W$ 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 $\Pi$ of ${\rm GSp}_2(\A)$ related to $g$ and an explicit vector $E_Z$ sits inside $\Pi$ which creates a vector valued (non-cuspidal) Siegel modular form of weight $(3,1)$ so that $F_Z$ coincides with $E_Z$ in $H^{2,0}(\partial W)$ under the Poincar\'e residue map and various identifications of cohomologies.