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Arithmetic properties of Picard-Fuchs equations and holonomic recurrences. (English) Zbl 1319.11023

Summary: 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 \(\mathbb Z[t]\). 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 \(\varGamma_1(7)\). We conclude with a consideration of asymptotics, wherein it is proved that many coefficient series satisfy asymptotic expressions of the form \(u_n\sim\ell\lambda^n/n\). Certain arithmetic results extend to the study of general holonomic recurrences.

MSC:

11F03 Modular and automorphic functions
11B37 Recurrences
11B83 Special sequences and polynomials
14H52 Elliptic curves
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