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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.

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)
Full Text: DOI
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