Eta-quotients and elliptic curves. (English) Zbl 0894.11020

Let \(\eta\) denote the Dedekind eta-function. Let positive integers \(s\), \(t_1,\dots, t_s\) and integers \(r_1,\dots, r_s\) be given. Then \[ \eta(t_1z)^{r_1} \eta(t_2z)^{r_2}\dots \eta(t_sz)^{r_s} \] is called an eta-quotient. Recently Y. Martin has shown [Trans. Am. Math. Soc. 348, 4825-4856 (1996; Zbl 0872.11026)] that there are exactly 74 eta-quotients of integral weight such that both this function and its transform under the Fricke involution are Hecke eigenforms. Now the authors show that exactly 12 of these eta-quotients are newforms of weight 2. For each of them, they give a modular elliptic curve \(E\) over \(\mathbb{Q}\) whose Hasse-Weil \(L\)-function \(L(E,s)\) agrees with the Mellin transform of that newform. Exactly 5 of these curves have complex multiplication. This implies that \(L(E,s)\) is a Hecke \(L\)-function with grössencharacter. Classical \(\eta\)-identities (Euler, Jacobi) are used to find, in these 5 cases, simple explicit formulae for the coefficients of the newforms at primes \(p\).


11F20 Dedekind eta function, Dedekind sums
11G05 Elliptic curves over global fields
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F11 Holomorphic modular forms of integral weight


Zbl 0872.11026


Full Text: DOI


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