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On the zeta function of a family of quintics. (English) Zbl 1217.11063
Summary: We give a proof of the link between the zeta function of two families of hypergeometric curves and the zeta function of a family of quintics that was observed numerically by P. Candelas, X. de la Ossa and F. Rodriguez Villegas [Calabi-Yau manifolds over finite fields. I. arXiv:hep-th/0012233, II in: Calabi-Yau varieties and mirror symmetry. Providence, RI: American Mathematical Society (AMS). Fields Inst. Commun. 38, 121–157 (2003; Zbl 1100.14032)]. The method we use is based on formulas of Koblitz and various Gauss sums identities; it does not give any geometric information on the link.

MSC:
11G42 Arithmetic mirror symmetry
11G25 Varieties over finite and local fields
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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