Rédei's triple symbols and modular forms

Fumiya Amano, Hisatoshi Kodani, Masanori Morishita, Takayuki Sakamoto, Takafumi Yoshida, Takeshi Ogaswara

Research output: Contribution to journalReview article

1 Citation (Scopus)

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 [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.

Original languageEnglish
Pages (from-to)405-427
Number of pages23
JournalTokyo Journal of Mathematics
Volume36
Issue number2
DOIs
Publication statusPublished - Dec 1 2013

Fingerprint

Modular Forms
Fourier coefficients
Prime number
Binary Quadratic Forms
Reciprocity Law
Theta Series
Legendre
Gauss
Genus
Decompose
Generalise
Theorem

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Amano, F., Kodani, H., Morishita, M., Sakamoto, T., Yoshida, T., & Ogaswara, T. (2013). Rédei's triple symbols and modular forms. 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.

In: Tokyo Journal of Mathematics, Vol. 36, No. 2, 01.12.2013, p. 405-427.

Research output: Contribution to journalReview article

Amano, F, Kodani, H, Morishita, M, Sakamoto, T, Yoshida, T & Ogaswara, T 2013, 'Rédei's triple symbols and modular forms', Tokyo Journal of Mathematics, vol. 36, no. 2, pp. 405-427. https://doi.org/10.3836/tjm/1391177979
Amano F, Kodani H, Morishita M, Sakamoto T, Yoshida T, Ogaswara T. Rédei's triple symbols and modular forms. Tokyo Journal of Mathematics. 2013 Dec 1;36(2):405-427. https://doi.org/10.3836/tjm/1391177979
Amano, Fumiya ; Kodani, Hisatoshi ; Morishita, Masanori ; Sakamoto, Takayuki ; Yoshida, Takafumi ; Ogaswara, Takeshi. / Rédei's triple symbols and modular forms. In: Tokyo Journal of Mathematics. 2013 ; Vol. 36, No. 2. pp. 405-427.
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