Arithmetic Properties of Picard-Fuchs Equations and Holonomic Recurrences

TL;DR

This paper explores the number-theoretic properties of Picard-Fuchs equations related to elliptic curves, demonstrating integrality conditions, congruences, and asymptotic behaviors of their solutions, especially for specific congruence subgroups.

Contribution

It extends the understanding of Picard-Fuchs solutions for general elliptic families, providing integrality criteria, congruence relations, and asymptotic formulas, especially for index 24 and 7 subgroups.

Findings

Picard-Fuchs solutions can be made integral via reparametrization.

A congruence relation similar to Atkin-Swinnerton-Dyer is established for mma_1(7).

Coefficient series exhibit specific asymptotic growth patterns.

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 \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.