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

Takeo Okazaki, Takuya Yamauchi

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1 Citation (Scopus)


Let Z be the quotient of the Siegel modular threefold Asa(2, 4, 8) which has been studied by van Geemen and Nygaard. They gave an implication that some 6-tuple FZ 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 EZ sits inside π which creates a vector valued (non-cuspidal) Siegel modular form of weight (3, 1) so that FZ coincides with EZ in H2,0(∂W) under the Poincaré residue map and various identifications of cohomologies.

Original languageEnglish
Pages (from-to)585-630
Number of pages46
JournalAdvances in Theoretical and Mathematical Physics
Issue number3
Publication statusPublished - 2017 Jan 1

ASJC Scopus subject areas

  • Mathematics(all)
  • Physics and Astronomy(all)


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