Theta Series Of Quadratic Forms Research Articles

A proof of Hecke’s formula for binary quadratic forms

In Mathematische Werke, Hecke defines the operator [Formula: see text] and describes their utility in conjunction with theta series of quadratic forms. In particular, he shows that the image of theta series associated to classes of binary quadratic forms in CL[Formula: see text] is again a theta series associated to a collection of forms in CL[Formula: see text]. We state and prove an explicit formula for the action of [Formula: see text] on a binary quadratic form of negative discriminant.

Shimura lifting of modular forms of weight 3/2

We show that there is the Shimura lifting map of \(\mathbf {M}_{3/2}(N,\chi _{0})\) to \(\mathbf {M}_{2}(N/2,\chi _{0}^{2})\) for any natural number N divisible by 4 and for any odd Dirichlet character \(\chi _{0}\), which was already proved in the case of higher weight (Tsuyumine, Tsukuba J Math 23:465–483, 1999). As the application, we show that the lift of the weighted average of theta series of quadratic forms of an odd number of variables in a class, is an Eisenstein series of integral weight.