Theta functions of quadratic forms over imaginary quadratic fields

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 ${\displaystyle {\vartheta }_Q\left(z\right):={\sum }_{g\in {\mathbb{Z}}^n}\exp \{\pi i\{^tg\}Qgz\}}$, Im z > 0, is a modular form of weight n/2 on ${\displaystyle {\Gamma }_0\left(N\right)}$, where N is the level of Q, i.e. ${\displaystyle N{Q}^{-1}}$ is integral and ${\displaystyle N{Q}^{-1}}$ 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. Stark [8] gives a different proof by converting ${\displaystyle {\vartheta }_Q\left(z\right)}$ into a symplectic theta function and then using the transformation formula for the symplectic theta function. In [4], we apply Stark's method and use theta functions of indefinite quadratic forms to construct modular forms over totally real number fields. In this paper, we define theta functions attached to quadratic forms over imaginary quadratic fields. We show that these theta functions are modular forms of weight n/2 on some ${\displaystyle {\Gamma }_0}$ groups by regarding them as symplectic theta functions and then applying well known results for symplectic theta functions. In particular, the main result of [8] allows us to compute the theta multiplier for our theta functions in a very elegant way.