Congruence properties of Taylor coefficients of modular forms

2014/06/11 by Hannah Larson, Geoffrey Smith, Larson, Hannah +1
Mathematics · #11F11 #11F33 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1406.2999

openalex publication_date 2014/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 τ. 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 τ 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^th Taylor coefficient of f at τ that are sharp under certain additional hypotheses.

Citations

• Modular Elliptic Curves and Fermat's Last Theorem

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