Congruence properties of Taylor coefficients of modular forms

TL;DR

This paper investigates the congruence properties of Taylor coefficients of modular forms expanded around CM points, revealing conditions under which prime powers divide these algebraic parts and providing bounds on such divisibility.

Contribution

It extends the study of Fourier coefficient congruences to Taylor coefficients at CM points, establishing divisibility conditions and effective bounds.

Findings

Prime power divisibility of Taylor coefficients under certain conditions

Effective bounds on the largest index with non-divisible coefficients

Conditions relating CM points, primes, and divisibility properties

Abstract

In their work, Serre and Swinnerton-Dyer study the congruence properties of the Fourier coefficients of modular forms. We examine similar congruence properties, but for the coefficients of a modified Taylor expansion about a CM point $\tau$. These coefficients can be shown to be the product of a power of a constant transcendental factor and an algebraic integer. In our work, we give conditions on $\tau$ and a prime number $p$ that, if satisfied, imply that $p^m$ divides the algebraic part of all the Taylor coefficients of $f$ of sufficiently high degree. We also give effective bounds on the largest $n$ such that $p^m$ does not divide the algebraic part of the $n^{\text{th}}$ Taylor coefficient of $f$ at $\tau$ that are sharp under certain additional hypotheses.