Eta-quotients of prime or semiprime level and elliptic curves

Allen, Michael; Anderson, Nicholas; Hamakiotes, Asimina; Oltsik, Ben; Swisher, Holly

doi:10.2140/involve.2020.13.879

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Eta-quotients of prime or semiprime level and elliptic curves

Michael Allen, Nicholas Anderson, Asimina Hamakiotes, Ben Oltsik and Holly Swisher

Vol. 13 (2020), No. 5, 879–900

DOI: 10.2140/involve.2020.13.879

Abstract

From the modularity theorem proven by Wiles, Taylor, Conrad, Diamond, and Breuil, we know that all elliptic curves are modular. It has been shown by Martin and Ono exactly which are represented by eta-quotients, and some examples of elliptic curves represented by modular forms that are linear combinations of eta-quotients have been given by Pathakjee, RosnBrick, and Yoong.

In this paper, we first show that eta-quotients which are modular for any congruence subgroup of level $N$ coprime to $6$ can be viewed as modular for $Γ_{0} (N)$ . We then categorize when even-weight eta-quotients can exist in $M_{k} (Γ_{1} (p))$ and $M_{k} (Γ_{1} (p q))$ for distinct primes $p, q$ . We conclude by providing some new examples of elliptic curves whose corresponding modular forms can be written as a linear combination of eta-quotients, and describe an algorithmic method for finding additional examples.

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Keywords

eta-quotients, modular forms, elliptic curves

Mathematical Subject Classification

Primary: 11F20, 11F37

Secondary: 11G05

Milestones

Received: 4 August 2020

Accepted: 11 August 2020

Published: 5 December 2020

Communicated by Ken Ono

Authors

Michael Allen
Oregon State University Corvallis, OR United States

Nicholas Anderson
Queen Mary University of London London United Kingdom

Asimina Hamakiotes
University of Connecticut Storrs, CT United States

Ben Oltsik
University of Connecticut Storrs, CT United States

Holly Swisher
Department of Mathematics Oregon State University Corvallis, OR United States

Vol. 13, No. 5, 2020