  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 \Gamma_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.
