Counting eta-quotients of prime level

Arnold-Roksandich, Allison; James, Kevin; Keaton, Rodney

doi:10.2140/involve.2018.11.827

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Counting eta-quotients of prime level

Allison Arnold-Roksandich, Kevin James and Rodney Keaton

Vol. 11 (2018), No. 5, 827–844

DOI: 10.2140/involve.2018.11.827

Abstract

It is known that a modular form on ${SL}_{2} (ℤ)$ can be expressed as a rational function in $η (z)$ , $η (2 z)$ and $η (4 z)$ . By using known theorems 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 a subspace spanned by the eta-quotients. In this paper, we primarily focus on the case where the level is $N = p$ , a prime. In this case, we will show an explicit count for the number of eta-quotients of level $p$ and show that they are linearly independent.

Keywords

modular forms, eta-quotients, Dedekind eta-function, number theory

Mathematical Subject Classification 2010

Primary: 11F11, 11F20, 11F37

Milestones

Received: 18 December 2016

Revised: 30 July 2017

Accepted: 24 August 2017

Published: 2 April 2018

Communicated by Kenneth S. Berenhaut

Authors

Allison Arnold-Roksandich
Department of Mathematics Oregon State University Corvallis, OR United States

Kevin James
Department of Mathematical Sciences Clemson University Clemson, SC United States

Rodney Keaton
Department of Mathematics and Statistics East Tennessee State University Johnson City, TN United States

Vol. 11, No. 5, 2018