Higher reciprocity law, modular forms of weight 1 and elliptic curves.(English)Zbl 0569.12007

Let $$f(x)=ax^ 3+bx^ 2+cx+d\in {\mathbb{Q}}[x]$$ be an irreducible polynomial over the rationals whose splitting field $$K_ f$$ is a Galois extension over $${\mathbb{Q}}$$ with Galois group $$G(K/{\mathbb{Q}})\cong S_ 3$$. The well-known connection between the higher reciprocity law of f(x) and modular functions of weight one [see also S. Chowla and M. Cowles, J. Reine Angew. Math. 292, 115-116 (1977; Zbl 0347.10011) and T. Hiramatsu, Comment. Math. Univ. St. Pauli 31, 75-85 (1982; Zbl 0494.10015)] is studied.
The connection between reciprocity laws and the theory of elliptic curves whose 2-torsion generate an extension of $${\mathbb{Q}}$$ with Galois group isomorphic to $$S_ 3$$, is also studied. Similar ideas to the present paper have also been considered in H. Ito [J. Reine Angew. Math. 332, 151-155 (1982; Zbl 0491.14015)] and overlooked by the author.