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The Eichler Commutation Relation for theta series with spherical harmonics

Lynne H. Walling (1993)

Acta Arithmetica

It is well known that classical theta series which are attached to positive definite rational quadratic forms yield elliptic modular forms, and linear combinations of theta series attached to lattices in a fixed genus can yield both cusp forms and Eisenstein series whose weight is one-half the rank of the quadratic form. In contrast, generalized theta series - those augmented with a spherical harmonic polynomial - will always yield cusp forms whose weight is increased by the degree of the...

Theta functions of quadratic forms over imaginary quadratic fields

Olav K. Richter (2000)

Acta Arithmetica

1. Introduction. Let Q be a positive definite n × n matrix with integral entries and even diagonal entries. It is well known that the theta function ϑ Q ( z ) : = g n e x p π i t g Q g z , Im z > 0, is a modular form of weight n/2 on Γ 0 ( N ) , where N is the level of Q, i.e. N Q - 1 is integral and N Q - 1 has even diagonal entries. This was proved by Schoeneberg [5] for even n and by Pfetzer [3] for odd n. Shimura [6] uses the Poisson summation formula to generalize their results for arbitrary n and he also computes the theta multiplier explicitly....

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