Weakly holomorphic modular forms in prime power levels of genus zero

2017/03/23 by Jenkins, Paul, Thornton, DJ
#11F33 #11F37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1703.08145

Abstract

Let M_k^\sharp(N) be the space of weight k, level N weakly holomorphic modular forms with poles only at the cusp at ∞. We explicitly construct a canonical basis for M_k^\sharp(N) for N∈\8,9,16,25\, and show that many of the Fourier coefficients of the basis elements in M₀^\sharp(N) are divisible by high powers of the prime dividing the level N. Additionally, we show that these basis elements satisfy a Zagier duality property, and extend Griffin's results on congruences in level 1 to levels 2, 3, 4, 5, 7, 8, 9, 16, and 25.

Related

• Divisibility properties of coefficients of modular functions in genus zero levels
• Zagier duality for level p weakly holomorphic modular forms
• Zeros of weakly holomorphic modular forms of levels 2 and 3
• Zeros of weakly holomorphic modular forms of level 4
• Congruences for coefficients of modular functions