Arithmetic properties of Picard–Fuchs equations and holonomic recurrences

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〛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)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/nun∼ℓλn/n. Certain arithmetic results extend to the study of general holonomic recurrences.