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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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