A Remark on Differences of Theta Series Original Research Article

The Eichler Commutation Relation shows that the space spanned by theta series attached to lattices in a given family (a finite collection of genera) is invariant under a particular subalgebra of the Hecke algebra. In previous work the author used this relation to construct eigenforms for this subalgebra; the magnitude of the eigenvalues shows these eigenforms are in fact Eisenstein series. In this paper we generalize a result of Siegel, showing that the difference of theta series attached to lattices in the same genus is a cusp form. We conclude that the space of theta series for a given family splits as a direct sum of the space spanned by the previously constructed eigenforms and a space of cusp forms.