## 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 $$p$$.

### MSC:

 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

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### References:

 [1] J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. · Zbl 0758.14042 [2] D. Dummit, H. Kisilevsky, and J. McKay, Multiplicative properties of $$\eta$$-functions, Contemp. Math. 45, Amer. Math. Soc. (1985), 89-98. · Zbl 0578.10028 [3] Basil Gordon and Dale Sinor, Multiplicative properties of \?-products, Number theory, Madras 1987, Lecture Notes in Math., vol. 1395, Springer, Berlin, 1989, pp. 173 – 200. · Zbl 0688.10023 [4] Basil Gordon and Sinai Robins, Lacunarity of Dedekind \?-products, Glasgow Math. J. 37 (1995), no. 1, 1 – 14. · Zbl 0817.11027 [5] Basil Gordon and Kim Hughes, Multiplicative properties of \?-products. II, A tribute to Emil Grosswald: number theory and related analysis, Contemp. Math., vol. 143, Amer. Math. Soc., Providence, RI, 1993, pp. 415 – 430. · Zbl 0808.11030 [6] Neal Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1984. · Zbl 0553.10019 [7] G. Köhler, Theta series on the theta group, Abh. Math. Sem. Univ. Hamburg 58 (1988), 15 – 45. · Zbl 0652.10020 [8] Günter Köhler, Theta series on the Hecke groups \?(\sqrt 2) and \?(\sqrt 3), Math. Z. 197 (1988), no. 1, 69 – 96. · Zbl 0632.10024 [9] I. G. Macdonald, Affine root systems and Dedekind’s \?-function, Invent. Math. 15 (1972), 91 – 143. · Zbl 0244.17005 [10] Yves Martin, Multiplicative \?-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825 – 4856. · Zbl 0872.11026 [11] Yves Martin, On Hecke operators and products of the Dedekind \?-function, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 4, 307 – 312 (English, with English and French summaries). · Zbl 0847.11019 [12] Geoffrey Mason, \?$$_{2}$$$$_{4}$$ and certain automorphic forms, Finite groups — coming of age (Montreal, Que., 1982) Contemp. Math., vol. 45, Amer. Math. Soc., Providence, RI, 1985, pp. 223 – 244. [13] Geoffrey Mason, On a system of elliptic modular forms attached to the large Mathieu group, Nagoya Math. J. 118 (1990), 177 – 193. · Zbl 0709.11033 [14] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1992. Corrected reprint of the 1986 original. · Zbl 0585.14026 [15] Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. · Zbl 0911.14015
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