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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.
Reviewer: J.A.Antoniadis

MSC:

11R39 Langlands-Weil conjectures, nonabelian class field theory
11F11 Holomorphic modular forms of integral weight
11F80 Galois representations
14H05 Algebraic functions and function fields in algebraic geometry
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References:

[1] Comm. Math. Univ. St. Paul 31 pp 75– (1982)
[2] J. reine angew. Math 292 pp 115– (1977)
[3] The modular equation and modular forms of weight one · Zbl 0557.10022
[4] On indefinite modular forms of weight one · Zbl 0587.12005
[5] DOI: 10.1016/0022-314X(80)90073-6 · Zbl 0426.10024
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