Eichler commutation relations for theta series of nebentypus
From the work of Eichler we have the fundamental principle that the effect of a Hecke operator \(T(m)\) acting on a suitable set of elliptic modular theta series can be represented by an ``Anzahlmatrix'' provided \(m\) is the norm of a similitude of the underlying quadratic space. If \(m\) is not the norm of a similitude no such ``Eichler commutation relations'' are known. We consider the case of theta series of weight two, prime level \(p\equiv 1\pmod 4\) and quadratic character \((\cdot/p)\). These come from positive definite integral quaternary quadratic forms of discriminant \(p\) and their adjoints of discriminant \(p^3\). In this situation we have an Eichler commutation relation for \(T(m)\) as long as \(m\) is a norm from \(K=\mathbb{Q}(\sqrt{p})\). The Anzahlmatrices can be expressed in terms of Brandt matrices associated to the definite quaternion algebra of discriminant 1 over \(K\). In addition, we have the standard action of \(W=\left[ \begin{matrix} 0 & -1\\ p & 0\end{matrix} \right]\) corresponding to the theta inversion formula. We consider the question of whether the resulting ``Brandt matrix representation'' can be extended to the algebra generated by \(W\) and all Hecke operators. Theorem 1 states that a necessary condition for such an extension to exist is that \(\pm \sqrt{p}\) not be an eigenvalue of \(T(p)\). Theorem 2 gives a partial converse to Theorem 1: The representation is extendable if \(\pm 2\sqrt{p}\) is not an eigenvalue of \(B(\mathfrak p)\), the Brandt matrix associated to the ramified prime \(\mathfrak p\) of \(K\). We note that the condition in Theorem 1 is also Waldspurger's criterion for a positive solution to the basis problem.
scientific article; zbMATH DE number 4019154