Weakly holomorphic modular forms in prime power levels of genus zero

Paul Jenkins; DJ Thornton

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arXiv · 1703.08145

Weakly holomorphic modular forms in prime power levels of genus zero

Paul Jenkins·DJ Thornton

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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 $\infty$. We explicitly construct a canonical basis for $M_k^\sharp(N)$ for $N\in\{8,9,16,25\}$, and show that many of the Fourier coefficients of the basis elements in $M_0^\sharp(N)$ 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.

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Paul Jenkins, DJ Thornton. 2017-03-23. Weakly holomorphic modular forms in prime power levels of genus zero. https://arxiv.org/abs/1703.08145

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