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arXiv · 1811.07244

Proceedings Paper for REU Project Involving Counting Eta-Quotients

Allison Arnold-Roksandich·Kevin James·Rodney Keaton
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Abstract

It is known that all modular forms on $SL_2(Z)$ can be expressed as a rational function in $\eta(z)$, $\eta(2z)$ and $\eta(4z)$. By using a theorem by Gordon, Hughes, and Newman, and calculating the order of vanishing, we can compute the $\eta$-quotients for a given level. Using this count, knowing how many $\eta$-quotients are linearly independent and using the dimension formula, we can figure out how the $\eta$-quotients span higher levels. In this paper, we primarily focus on the case where $N=p$ a prime, and some discussion for non-prime indicies.

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Allison Arnold-Roksandich, Kevin James, Rodney Keaton. 2018-11-17. Proceedings Paper for REU Project Involving Counting Eta-Quotients. https://arxiv.org/abs/1811.07244

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