Eta-quotients of Prime or Semiprime Level and Elliptic Curves

Abstract

From the Modularity Theorem proven by Wiles, Taylor, et al, 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 M k (Γ1(p)) and M k (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.

Existence of $η$-Quotients For Squarefree Levels

“…Additional work on spaces of modular forms spanned by η-quotients has been done by Arnold-Roksandich, James, and Keaton [2] and Rouse and Webb [12], for example. In [1], the author et al investigate the question of which modular form spaces cannot contain η-quotients; these results are extended in this paper.…”

“…where τ is in the upper-half of the complex plane and q := e 2πiτ , is one of the most famous and classical examples of a weight 1 2 modular form. The infinite product formula makes η(τ ) particularly nice to work with for many computations.…”

“…Additionally, the converse to Theorem 1.1 holds when N is coprime to 6. Moreover, any weakly holomorphic η-quotient on Γ 1 (N) must also satisfy the hypotheses of Theorem 1.1 (see Corollary 1.8 of [1]). That is, given…”

“…In [1], the author et al investigate other factors in addition to Theorem 1.3 which inhibit the existence of η-quotients in certain modular form spaces. The first result from [1] gives a complete categorization of which space M k (p) contain η-quotients when k ≥ 2 is even and p > 5 is prime.…”

“…Theorem 1.4 (Theorem 1.9 [1]). Let p be prime, set h p = 1 2 gcd(p − 1, 24), and let k be an even integer.…”