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On the transcendence degree of the differential field generated by Siegel modular forms. (English) Zbl 1130.11020
Let $$q$$ be a positive integer and $$k$$ be an algebraically closed subfield of $$\mathbb{C}$$, $$\mathfrak{H}_g$$ Siegel half space of degree $$g$$, $$\tau = (\tau_{jl})$$ a generic point on $$\mathfrak{H}_g$$, and $$\Gamma$$ a congruence subgroup of the symplectic group $$\mathrm{Sp}_{2g}K$$. The field of modular functions $$K=K(\Gamma, k)$$ has the form $$K=k(\lambda)$$, where $$\lambda=\{\lambda_1, \ldots, \lambda_N\}$$ is a set of modular function relative to $$\Gamma$$, whose first $$n=\mathrm{tr deg}(K/k)$$ elements are algebraically independent. Set $${\delta} =\{\delta_{jl}, 1\leq j \leq l \leq q \}$$ where $$\delta_{ij} =\frac{1}{2\pi i}\frac{\partial}{\partial \tau_{jl}}, 1\leq j< l <g, \delta_{jj} =\frac{1}{\pi i} \frac{\partial}{\partial\tau_{jj}}, 1\leq j \leq g$$. Let $$M=M(\Gamma, k)=k\langle \lambda_1, \ldots, \lambda_N \rangle$$ be the $${\delta}$$-differential field generated by $$K$$.
The main result of the paper (Theorem 1) asserts that $$M$$ is a finite extension of the field generated over $$K$$ by the $${\delta}$$-partial derivatives $$\lambda_1, \ldots, \lambda_N$$ of order $$\leq 2$$ and has the transcendence degree $$2g^2 +g$$, $$M$$ and $$\mathbb{C}({\tau})$$ are linearly disjoint over $$k$$, and $$\mathrm{tr deg} (M(2 \pi i \tau)/h)=\frac{g(5g+3)}{2}$$.
Taking for $$\Gamma$$ the theta group $$\Gamma_{4,8}$$ of level 4,8, and $$\lambda=\{ \vartheta_a / \vartheta_0 , a\in (\mathbb{Z}/2\mathbb{Z})^{2g}\}$$ this result can be stated in the following more precise theorem:
Theorem 2. The $$\delta$$-derivative of order $$\leq 2$$ of the modular functions $$\{ \vartheta_a / \vartheta_0 , a\in (\mathbb{Z}/2\mathbb{Z})^{2g}\}$$ generate over $$k$$ a $$\delta$$-stable field $$M(\Gamma_{4,8}, k)$$ of transcendence degree $$2g^2+g$$ over $$k$$, over which $$\vartheta_0$$ is algebraic. F
inally, this last result is sharpened in the cases $$g=1,2$$. This is done in the final section of the paper. The proofs use the Picard-Fuchs and Picard-Vessiot theories, differential forms and the explicit formulae on derivatives of theta functions.

MSC:
 11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 12H05 Differential algebra
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