Congruences for coefficients of modular functions

2014/04/02 by Paul M. Jenkins, Jenkins, Paul, DJ Thornton +1
Mathematics · #11F30 #11F33 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1404.0699

openalex publication_date 2014/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We examine canonical bases for weakly holomorphic modular forms of weight 0 and level p = 2, 3, 5, 7, 13 with poles only at the cusp at ∞. We show that many of the Fourier coefficients for elements of these canonical bases are divisible by high powers of p, extending results of the first author and Andersen. Additionally, we prove similar congruences for elements of a canonical basis for the space of modular functions of level 4, and give congruences modulo arbitrary primes for coefficients of such modular functions in levels 1, 2, 3, 4, 5, 7, and 13.

Related

• Non-ordinary primes and congruences for weakly holomorphic modular forms of level 2
• Congruences for coefficients of modular functions in levels 3, 5, and 7\n with poles at 0
• Weakly holomorphic modular forms in prime power levels of genus zero
• Divisibility properties of coefficients of modular functions in genus zero levels
• Congruences for coefficients of level 2 modular functions with poles at 0