A Siegel modular threefold and Saito-Kurokawa type lift to \(S_{3}(\Gamma_{1,3}(2))\)
The compactified Siegel modular threefold associated to the group \(\Gamma_{1,3}(2)\) (the moduli space of \((1,3)\)-polarised abelian surfaces with level-\(2\) structure) was shown to be a Calabi-Yau by \textit{W. Barth} and \textit{I. Nieto} [J. Algebr. Geom. 3, No. 2, 173--222 (1994; Zbl 0809.14027)] using geometric methods. The Siegel modular form \(F\) that gives the nowhere vanishing \(3\)-form was found by \textit{V. Gritsenko} and \textit{K. Hulek} [Int. Math. Res. Not. 1999, No. 17, 915--937 (1999; Zbl 0943.11027)], who described it as a Borcherds form. It was later conjectured [\textit{K. Hulek, J. Spandaw, B. van Geemen} and \textit{D. van Straten}, Adv. Geom. 1, No. 3, 263--289 (2001; Zbl 1014.14021)] that \(F\) comes from a Saito-Kurokawa lift, and in particular that the spinor \(L\)-function of \(F\) is \[ \zeta(s-1)\zeta(s-2)L(s,h) \] where \(h\) is the newform of weight~\(4\) for \(\Gamma_0(6)\). In this paper the authors construct \(F\) in that way. They also conjecture that the same should be true for any Siegel modular threefold~\(X\) over \({\mathbb Q}\) with \(h^{3,0}(X)=1\), \(h^{2,1}(X)=0\). The reviewer is not aware of any other such varieties, though; and the authors also ask for examples with \(h^{3,0}(X)=0\), \(h^{2,1}(X)=1\) (the reviewer is not aware of any of those either). The version of the Saito-Kurokawa lift that is used here is \textit{H. Yoshida}'s [Invent. Math. 60, 193--248 (1980; Zbl 0453.10022)] and the paper follows the procedure laid out there. A particular difficulty is that the most obvious choice for the Schwarz function on a subset \(H_{\mathbb A}\) of \(B_{\mathbb A}^2\) (where \(B_{\mathbb Q}\) is a suitably ramified quaternion algebra), needed to temper the integral over \(H_{\mathbb Q}\backslash H_{\mathbb A}\) that defines the lift, does not have the right properties at the primes \(2\) and \(3\). Therefore the authors give a customised Schwarz function, modified at those primes so as to behave as desired.
scientific article; zbMATH DE number 5284758