### 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 |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Tokyo Journal of Mathematics*,

*36*(2), 405-427. https://doi.org/10.3836/tjm/1391177979

**Rédei's triple symbols and modular forms.** / Amano, Fumiya; Kodani, Hisatoshi; Morishita, Masanori; Sakamoto, Takayuki; Yoshida, Takafumi; Ogaswara, Takeshi.

Research output: Contribution to journal › Review article

*Tokyo Journal of Mathematics*, vol. 36, no. 2, pp. 405-427. https://doi.org/10.3836/tjm/1391177979

}

TY - JOUR

T1 - Rédei's triple symbols and modular forms

AU - Amano, Fumiya

AU - Kodani, Hisatoshi

AU - Morishita, Masanori

AU - Sakamoto, Takayuki

AU - Yoshida, Takafumi

AU - Ogaswara, Takeshi

PY - 2013/12/1

Y1 - 2013/12/1

N2 - In 1939, L. Rédei introduced a certain triple symbol in order to generalize the Legendre symbol and Gauss' genus theory. Rédei's triple symbol [a1,a2, p] describes the decomposition law of a prime number p in a certain dihedral extension over Q of degree 8 determined by a1 and a2. In this paper, we show that the triple symbol [-p1, P2, P3] for certain prime numbers p 1, p2 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.

AB - In 1939, L. Rédei introduced a certain triple symbol in order to generalize the Legendre symbol and Gauss' genus theory. Rédei's triple symbol [a1,a2, p] describes the decomposition law of a prime number p in a certain dihedral extension over Q of degree 8 determined by a1 and a2. In this paper, we show that the triple symbol [-p1, P2, P3] for certain prime numbers p 1, p2 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.

UR - http://www.scopus.com/inward/record.url?scp=84897658238&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84897658238&partnerID=8YFLogxK

U2 - 10.3836/tjm/1391177979

DO - 10.3836/tjm/1391177979

M3 - Review article

AN - SCOPUS:84897658238

VL - 36

SP - 405

EP - 427

JO - Tokyo Journal of Mathematics

JF - Tokyo Journal of Mathematics

SN - 0387-3870

IS - 2

ER -