Redei's Triple Symbols and Modular Forms

In 1939, L. Redei introduced a certain triple symbol in order to generalize the Legendre symbol and Gauss' genus theory. Redei's triple symbol [a(1), a(2), p] describes the decomposition law of a prime number p in a certain dihedral extension over 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 Redei's dihedral extensions. A reciprocity law for the Redei triple symbols yields certain reciprocal relations among Fourier coefficients.