Congruence properties of Taylor coefficients of modular forms

Hannah Larson; Geoffrey Smith

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arXiv · 1406.2999

Congruence properties of Taylor coefficients of modular forms

Hannah Larson·Geoffrey Smith

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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.

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Hannah Larson, Geoffrey Smith. 2014-06-11. Congruence properties of Taylor coefficients of modular forms. https://arxiv.org/abs/1406.2999

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