## Abstract

In 1939, L. Rédei introduced a certain triple symbol in order to generalize the Legendre symbol and Gauss' genus theory. Rédei's triple symbol [a_{1},a_{2}, p] describes the decomposition law of a prime number p in a certain dihedral extension over Q of degree 8 determined by a_{1} and a_{2}. In this paper, we show that the triple symbol [-p_{1}, P_{2}, P_{3}] for certain prime numbers p _{1}, p_{2} and py can be expressed as a Fourier coefficient of a modular form of weight one. For this, we employ Hecke's theory on theta series associated to binary quadratic forms and realize an explicit version of the theorem by Weil-Langlands and Deligne-Serre for Rédei's dihedral extensions. A reciprocity law for the Rédei triple symbols yields certain reciprocal relations among Fourier coefficients.

Original language | English |
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Pages (from-to) | 405-427 |

Number of pages | 23 |

Journal | Tokyo Journal of Mathematics |

Volume | 36 |

Issue number | 2 |

DOIs | |

Publication status | Published - Dec 1 2013 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)