Die Nullstellen der Thetareihen zu positiven, binär-quadratischen Formen. (Zeros of theta series for positive binary quadratic forms)
Let \(Q\) be a primitive, positive definite, binary quadratic form with discriminant \(4N\), \(N\equiv 3\pmod 4\). For \(N\) small or \(Q\) diagonalizable the zeros of the associated \(\vartheta\)-function with characteristic \((\nu,\mu)\) on the upper half-plane \(\mathbb{H}\) is well-known; moreover for odd characteristics the \(\vartheta\)-function vanishes identically on \(\mathbb{H}\). In embedding \(\mathbb{H}\) in the Siegel-space \(\mathbb{H}_ 2\) and applying results on theta-constants of degree 2 the author shows for the product \(\Theta\) of the 10 \(\vartheta\)'s with even characteristic \(\Theta(Q,z):=\prod \vartheta(Q,(\nu,\mu))\) the following: (1) \(\Theta\) vanishes identically on \(\mathbb{H}\) if and only if \(Q\) is diagonalizable, (2) If \(Q\) is not diagonalizable \(\Theta\) has only simple zeros, which are characterized by a quadratic equation with integer coefficients. As an application new relations on class numbers for imaginary-quadratic number fields are found which in special cases trace back to the work of Eichler.
scientific article; zbMATH DE number 60090
scientific article