Tokyo Journal of Mathematics

Rédei's Triple Symbols and Modular Forms

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

Full-text: Open access


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

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

First available in Project Euclid: 31 January 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R
Secondary: 11F


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.

Export citation


  • F. Amano, On certain nilpotent extensions and multiple residue symbols, Master thesis, Kyushu University, 2012.
  • S. Arno, The imaginary quadratic fields of class number 4, Acta Arith. 60 (1992), no. 4, 321–334.
  • G. Fujisaki, Introduction to Algebraic Number Theory (Japanese), Shokabo, 1975.
  • E. Hecke, Zur Theorie der elliptischen Modulfunktionen, Math. Ann. 97 (1927), no. 1, 210–242.
  • T. Hiramatsu and Y. Mimura, The modular equation and modular forms of weight one, Nagoya Math. J. 100 (1985), 145–162.
  • T. Miyake, Modular Forms, Springer Monographs in Mathematics, Springer, 2006.
  • M. Morishita, On certain analogies between knots and primes, J. Reine Angew. Math. 550 (2002), 141–167.
  • T. Ono, An introduction to algebraic number theory, The University Series in Mathematics, Plenum Press, New York, 1990.
  • S. Saito, Number Theory (Japanese), Kyoritsu, 1997.
  • J.-P. Serre, Modular forms of weight one and Galois representations, In: Algebraic number fields$:$ L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Academic Press, London, 193–268.
  • L. Rédei, Ein neues zahlentheoretisches Symbol mit Anwendungen auf die Theorie der quadratischen Zahlkörper I, J. Reine Angew. Math. 180 (1939), 1–43.
  • L. Rozansky, Reshetikhin's formula for the Jones polynomial of a link: Feynman diagrams and Milnor's linking numbers, Topology and physics, J. Math. Phys. 35 (1994), no. 10, 5219–5246.
  • H. Wada and M. Saito, A Table of Ideal Class Groups of Imaginary Quadratic Fields, Sophia Kôkyûroku in Mathematics 28 (1988).
  • M. Watkins, Class numbers of imaginary quadratic fields, Math. Comp. 73 (2004), no. 246, 907–938.
  • D. Zagier, Elliptic modular forms and their application, In: The 1-2-3 of Modular Forms, Universitext, Springer, 1–103.