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Supercongruences for rigid hypergeometric Calabi-Yau threefolds. (English) Zbl 07436486
Summary: We establish the supercongruences for the fourteen rigid hypergeometric Calabi-Yau threefolds over \(\mathbb{Q}\) conjectured by Rodriguez-Villegas in 2003. Our first method is based on Dwork’s theory of \(p\)-adic unit roots and it allows us to establish the supercongruences between the truncated hypergeometric series and the corresponding unit roots for ordinary primes. The other method makes use of the theory of hypergeometric motives, in particular, adapts the techniques from the recent work of Beukers, Cohen and Mellit on finite hypergeometric sums over \(\mathbb{Q}\). Essential ingredients in executing the both approaches are the modularity of the underlying Calabi-Yau threefolds and a \(p\)-adic perturbation method applied to hypergeometric functions.

MSC:
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J33 Mirror symmetry (algebro-geometric aspects)
11F33 Congruences for modular and \(p\)-adic modular forms
11T24 Other character sums and Gauss sums
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
33C20 Generalized hypergeometric series, \({}_pF_q\)
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