Tokyo Journal of Mathematics

Rédei's Triple Symbols and Modular Forms

Fumiya AMANO, Hisatoshi KODANI, Masanori MORISHITA, Takeshi OGASAWARA, Takayuki SAKAMOTO, and Takafumi YOSHIDA

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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 $\mathbb{Q}$ 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 $p_3$ 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.

Article information

Source
Tokyo J. Math., Volume 36, Number 2 (2013), 405-427.

Dates
First available in Project Euclid: 31 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1391177979

Digital Object Identifier
doi:10.3836/tjm/1391177979

Mathematical Reviews number (MathSciNet)
MR3161566

Zentralblatt MATH identifier
1286.11052

Subjects
Primary: 11R
Secondary: 11F

Citation

AMANO, Fumiya; KODANI, Hisatoshi; MORISHITA, Masanori; SAKAMOTO, Takayuki; YOSHIDA, Takafumi; OGASAWARA, Takeshi. Rédei's Triple Symbols and Modular Forms. Tokyo J. Math. 36 (2013), no. 2, 405--427. doi:10.3836/tjm/1391177979. https://projecteuclid.org/euclid.tjm/1391177979


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