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Cusp forms of weight one, quartic reciprocity and elliptic curves

  • Noburo Ishii (a1)

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Let m be a non-square positive integer. Let K be the Galois extension over the rational number field Q generated by and . Then its Galois group over Q is the dihedral group D 4 of order 8 and has the unique two-dimensional irreducible complex representation ψ. In view of the theory of Hecke-Weil-Langlands, we know that ψ defines a cusp form of weight one (cf. Serre [6]).

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References

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[ 1 ] Davenport, H. and Hasse, H., Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fallen, J. reine u. angew. Math., 172 (1934), 151182.
[ 2 ] Hiramatsu, T., Higher reciprocity law and modular forms of weight one, Comm, Math. Univ. St. Paul., 31 (1982), 7585.
[ 3 ] Hiramatsu, T., Ishii, N. and Mimura, Y., On indefinite modular forms of weight one, preprint.
[ 4 ] Koike, M., Higher reciprocity law, modular forms of weight 1 and elliptic curves, Nagoya Math. J., 98 (1985),
[ 5 ] Moreno, C., The higher reciprocity law: an example, J. Number Theory, 12 (1980), 5770.
[ 6 ] Serre, J. P., Modular forms of weight one and Galois representations, Proc. Symposium on Algebraic Number Fields, Academic Press, London, 1977, 193268.
[ 7 ] Shimura, G., Introduction to the arithmetic theory of automorphic functions, Iwanami Shoten Publisher and Princeton Univ. Press, 1971.
[ 8 ] Shimura, G., On elliptic curves with complex multiplication as factors of the jacobians of modular function fields, Nagoya Math. J., 43 (1971), 199208.
[ 9 ] Tate, J., The arithmetic of elliptic curves, Invent. Math., 23 (1974), 179206.
[10] Tate, J., Algorithm for determining the type of a singular fiber in an elliptic pencil, Lecture Notes in Math., 476 (1975), 3352.
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Cusp forms of weight one, quartic reciprocity and elliptic curves

  • Noburo Ishii (a1)

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