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2020 Eta-quotients of prime or semiprime level and elliptic curves
Michael Allen, Nicholas Anderson, Asimina Hamakiotes, Ben Oltsik, Holly Swisher
Involve 13(5): 879-900 (2020). 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 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.

Citation

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Michael Allen. Nicholas Anderson. Asimina Hamakiotes. Ben Oltsik. Holly Swisher. "Eta-quotients of prime or semiprime level and elliptic curves." Involve 13 (5) 879 - 900, 2020. https://doi.org/10.2140/involve.2020.13.879

Information

Received: 4 August 2020; Accepted: 11 August 2020; Published: 2020
First available in Project Euclid: 22 December 2020

MathSciNet: MR4190445
Digital Object Identifier: 10.2140/involve.2020.13.879

Subjects:
Primary: 11F20, 11F37
Secondary: 11G05

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.13 • No. 5 • 2020
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