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A Hypergeometric Versionof the Modularity of Rigid Calabi–Yau Manifolds
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

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We examine instances of modularity of (rigid) Calabi–Yau manifolds whose periods are expressed in terms of hypergeometric functions. The $p$-th coefficients $a(p)$ of the corresponding modular form can be often read off, at least conjecturally, from the truncated partial sums of the underlying hypergeometric series modulo a power of $p$ and from Weil's general bounds $|a(p)|\le2p^{(m-1)/2}$, where $m$ is the weight of the form. Furthermore, the critical $L$-values of the modular form are predicted to be $\mathbb Q$-proportional to the values of a related basis of solutions to the hypergeometric differential equation.
Keywords: hypergeometric equation; bilateral hypergeometric series; modular form; Calabi–Yau manifold. 
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     author = {Wadim Zudilin},
     title = {A {Hypergeometric} {Versionof} the {Modularity} of {Rigid} {Calabi{\textendash}Yau} {Manifolds}},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a85/}
}
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Wadim Zudilin. A Hypergeometric Versionof the Modularity of Rigid Calabi–Yau Manifolds. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a85/

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