Rédei's triple symbols and modular forms

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

Research output: Contribution to journalReview articlepeer-review

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

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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