The structure of subnormal subgroups of symplectic groups over local rings.

Tazhetdinov, S.

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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 $ϑ_{Q} (z) : = \sum_{g \in ℤ^{n}} e x p π i^{t} g Q g z$ , 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...