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.
Citation
Fumiya AMANO. Hisatoshi KODANI. Masanori MORISHITA. Takeshi OGASAWARA. Takayuki SAKAMOTO. Takafumi YOSHIDA. "Rédei's Triple Symbols and Modular Forms." Tokyo J. Math. 36 (2) 405 - 427, December 2013. https://doi.org/10.3836/tjm/1391177979
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