Modularity of a nonrigid Calabi-Yau manifold with bad reduction at 13.

*(English)*Zbl 1114.14015The starting point of this paper is a small resolution \(\hat Y\) of the fiber product \(Y:={\mathcal{E}}_{(1:-3:-3)} \times {\mathcal{E}}^{Mt}(\Gamma(3))\) considered by K. Hulek and H. Verrill [in: Mirror Symmetry V, Adv. Stud. Math. 38, 19–34 (2006; Zbl 1115.14031)]. Here \({\mathcal{E}}(\Gamma(3)):=\{((t_0,t_1),(x,y,z))\in{\mathbb{P}}^1\times {\mathbb{P}}^2\,| \, (x^3+y^3+z^3)t_0=3xyzt_1\,\}\) and \({\mathcal{E}}_{(1:-3:-3)}:=\{((t_0,t_1),(x,y,z))\in{\mathbb{P}}^1\times{\mathbb{P}}^2\,| \, (x+y+z)(xy-3(xz+yz)t_0)=xyzt_1\,\}\) are elliptic fibrations, and \({\mathcal{E}}^{Mt}\) is the twist of \({\mathcal{E}}(\Gamma(3))\) by the automorphism \(Mt=(t+7)/8\). \(\hat Y\) is a nonrigid Calabi-Yau threefold with \(h^{2,1}=1\) defined over \({\mathbb Q}\) with bad reduction at \(13\). Thus the associated Galois representation is of dimension \(4\). Hulek and Verrill showed that the \(L\)-series of \(\hat Y\) is of the form \(L(\hat Y,s)=L(g_4,s)L(g_2,s-1)\) where \(g_2\) is a newform of weight \(2\) and level \(13\), and \(g_4\) is some newform of weight \(4\) and level \(27\). Based on numerical evidence, Hulek and Verrill predicted that \(g_4\) is \(27k4B\) in Stein’s notation [Modular forms database, http://modular.math.washington.edu].

The paper under review proves this prediction. The method used here is to consider an auxiliary Calabi-Yau threefold \(X\) constructed by replacing the fiber product by the Kummer fibration. Its smooth resolution \(\hat X\) is rigid and has good reduction at \(13\). The modularity of \(\hat X\) is known, that is, the \(L\)-series of \(\hat X\) is determined by the modular form \(27k4B\). Then using the fact that there is generically \(2:1\) correspondence \(\hat Y\;\to \hat X\), the modularity of \(\hat Y\) follows, establishing the Hulek–Verrill prediction.

The paper under review proves this prediction. The method used here is to consider an auxiliary Calabi-Yau threefold \(X\) constructed by replacing the fiber product by the Kummer fibration. Its smooth resolution \(\hat X\) is rigid and has good reduction at \(13\). The modularity of \(\hat X\) is known, that is, the \(L\)-series of \(\hat X\) is determined by the modular form \(27k4B\). Then using the fact that there is generically \(2:1\) correspondence \(\hat Y\;\to \hat X\), the modularity of \(\hat Y\) follows, establishing the Hulek–Verrill prediction.

Reviewer: Noriko Yui (Kingston)