Vol. 13, No. 5, 2020

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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

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 Γ0(N). We then categorize when even-weight eta-quotients can exist in Mk(Γ1(p)) and Mk(Γ1(pq)) 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.

eta-quotients, modular forms, elliptic curves
Mathematical Subject Classification
Primary: 11F20, 11F37
Secondary: 11G05
Received: 4 August 2020
Accepted: 11 August 2020
Published: 5 December 2020

Communicated by Ken Ono
Michael Allen
Oregon State University
Corvallis, OR
United States
Nicholas Anderson
Queen Mary University of 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