Eta-quotients of prime or semiprime level and elliptic curves

Existence of $$\eta $$-quotients for squarefree levels

“…Additional work on spaces of modular forms spanned by η-quotients has been done by Arnold-Roksandich et al [2] and Rouse and Webb [12], for example. In [1], the author et al investigate the question of which integer weight 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 2π iτ , 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 for example Remark 1.7 of [1]). That is, given f (τ ) = δ|N η(δτ ) r δ ∈ M !…”

“…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 ≥ 5 be prime, set h p = 1 2 gcd(p − 1, 24), and let k be an even integer.…”

“…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 ≥ 5 be prime, set h p = 1 2 gcd(p − 1, 24), and let k be an even integer. Then there exists f (τ ) = η(τ ) r 1 η(pτ ) r p ∈ M k (Γ 1 (p)) if and only if both of the following conditions hold.…”