On the linear independence of certain theta-series
The author gives a criterion for the linear independence of the Jacobi theta-series attached to even, positive definite, quaternary quadratic forms F of discriminant q (q an odd prime) representing 2 (i.e. there is an \(r\in {\mathbb{Z}}^ 4:\) \(F[r]=2)\) given by the formula \[ \vartheta_{F,r}(\tau,z)=\sum_{n\in {\mathbb{Z}}^ 4}\exp (2\pi i \tau F[n]+2\pi i zr^ tFn) \] (here \(\tau\in H\), the upper half-plane and \(z\in {\mathbb{C}})\). In this context, it is also possible to derive a criterion for the linear independence of the classical theta-series attached to the same quadratic forms. Further it is shown how these results are related to the criteria for the linear independence of the Jacobi theta-series attached to even, positive definite, quaternary quadratic forms of discriminant \(q^ 2\) representing 2 and their ''Nullwerte''. Especially for \(q=389\) all the theta-series in question are linearly dependent and satisfy one linear relation, which is explicitly given, answering a question of Gross about the linear relation satisfied by the theta-series attached to even, positive definite, ternary quadratic forms of discriminant \(32\cdot 389^ 2\).
scientific article; zbMATH DE number 3997904
scientific article