Congruence classes for modular forms over small sets

2023/02/06 by Subham Bhakta, S. Krishnamoorthy, Bhakta, Subham +3
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2302.02725

Abstract

J.P. Serre showed that for any integer m,~a(n)≡ 0 \pmod m for almost all n, where a(n) is the n^th Fourier coefficient of any modular form with rational coefficients. In this article, we consider a certain class of cuspforms and study #\a(n) \pmod m\_{n≤ x} over the set of integers with O(1) many prime factors. Moreover, we show that any residue class a∈ ℤ/mℤ can be written as the sum of at most thirteen Fourier coefficients, which are polynomially bounded as a function of m.

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• Non-vanishing Fourier coefficients of modular forms
• Fourier coefficients of half-integral weight modular forms modulo ell
• Congruence properties of Taylor coefficients of modular forms
• Distribution of integral Fourier Coefficients of a Modular Form of Half Integral Weight Modulo Primes
• Congruences for coefficients of modular functions