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- W2964309314 abstract "The coefficient series of the holomorphic Picard–Fuchs differential equation associated with the periods of elliptic curves often have surprising number-theoretic properties. These have been widely studied in the case of the torsion-free, genus zero congruence subgroups of index 6 and 12 (e.g. the Beauville families). Here, we consider arithmetic properties of the Picard–Fuchs solutions associated to general elliptic families, with a particular focus on the index 24 congruence subgroups. We prove that elliptic families with rational parameters admit linear reparametrizations such that their associated Picard–Fuchs solutions lie in Z〚t〛. A sufficient condition is given such that the same holds for holomorphic solutions at infinity. An Atkin–Swinnerton-Dyer congruence is proven for the coefficient series attached to Γ1(7). We conclude with a consideration of asymptotics, wherein it is proved that many coefficient series satisfy asymptotic expressions of the form un∼ℓλn/n. Certain arithmetic results extend to the study of general holonomic recurrences." @default.
- W2964309314 created "2019-07-30" @default.
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- W2964309314 date "2013-08-01" @default.
- W2964309314 modified "2023-10-03" @default.
- W2964309314 title "Arithmetic properties of Picard–Fuchs equations and holonomic recurrences" @default.
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- W2964309314 doi "https://doi.org/10.1016/j.jnt.2013.02.001" @default.
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