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# Advances in Theoretical and Mathematical Physics

## Volume 21 (2017)

### Number 3

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

Pages: 585 – 630

DOI: http://dx.doi.org/10.4310/ATMP.2017.v21.n3.a1

#### Authors

#### Abstract

Let $Z$ be the quotient of the Siegel modular threefold $\mathcal{A}^{\mathrm{sa}} (2, 4, 8)$ 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 $\mathrm{L}$-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 $\mathrm{L}$-function is described by some elliptic newform $g$ of weight $3$. Then we construct an automorphic representation $\Pi$ of $\mathrm{GSp}_2 (\mathbb{A})$ 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 Poincaré residue map and various identifications of cohomologies.

The first author is supported by JSPS Grant-in-Aid for Scientific Research No. 24740017.

The second author had been partially supported by JSPS Grant-in-Aid for Scientific Research No. 23740027 and JSPS Postdoctoral Fellowships for Research Abroad No. 378. He is now partially supported by JSPS Grant-in-Aid for Scientific Research (C) No. 15K04787.