On indefinite modular forms of weight one
Let us suppose that the ring class field \(N_ f\) modulo f (f\(\in {\mathbb{N}})\) of an imaginary quadratic field \(\Sigma\) is a dihedral extension over \({\mathbb{Q}}\) with Galois group \(D_ 4\). Let K be the unique real quadratic subfield of \(N_ f\). From a result of Shintani it follows that the L-series associated to \(\Sigma\) and to K (with special ring (ray) class characters \(\chi\) ' and \(\chi)\) coincide: \[ (1)\quad L_{\Sigma}(s,\chi ')=L_ K(s,\chi). \] The Mellin transform of the above L-series now gives an identity between the corresponding positive definite and indefinite theta series. This relation has been given explicitely in several numerical examples. At the end of the paper the higher reciprocity law on extensions of imaginary quadratic fields with Galoisgroup \(D_ 4\) [cf. \textit{C. J. Moreno}, J. Number Theory 12, 57-70 (1980; Zbl 0426.10024), and the reviewer, J. Reine Angew. Math. 361, 11- 22 (1985; Zbl 0562.12014)] is transfered to the case of real quadratic fields using equation (1).
scientific article; zbMATH DE number 3941642