A Decomposition Theorem on Differential Polynomials of Theta Functions of High Level

Let h and g be two positive integers. We fix an element Ω of the Siegel upper half plane H g := Z ∈ C (g,g) | Z = t Z, Im Z > 0 of degree g once and for all. Let M be positive symmetric, even integral matrix of degree h. An entire function f on C (h,g) satisfying the transformation behaviour F (W + ξΩ + η) = exp −πiσ(M(ξΩ t ξ + 2W t ξ)) f (W) for all W ∈ C (h,g) and (ξ, η) ∈ Z (h,g) × Z (h,g) is called a theta function of level M with respect to Ω. The set T M (Ω) of all theta functions of level M with respect to Ω forms a complex vector space of dimension (det M) g with a canonical basis consisting of theta series ϑ (M) A 0 (Ω|W) := N∈Z (h,g) exp πiσ(M((N + A)Ω t (N + A) + 2W t (N + A)))) , where A runs over a complete system of representatives of the cosets L M := M −1 Z (h,g) /Z (h,g). We let T (Ω) := M T M (Ω) be the graded algebra of theta functions, where M = (M kl) (1 ≤ k, l ≤ h) runs over the set M(h) of all positive symmetric, even integral h × h matrices with M kl = 0 for all k, l. In this paper we prove the following decomposition theorem: