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Modularity of a nonrigid Calabi-Yau manifold with bad reduction at 13. (English) Zbl 1114.14015
The 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.

MSC:
 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 11F23 Relations with algebraic geometry and topology
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