On the transcendence degree of the differential field generated by Siegel modular forms. (English) Zbl 1130.11020

Bertrand, D.; Zudilin, W.

J. Reine Angew. Math. 554, 47-68 (2003).

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.

Reviewer: Mykola Ya. Komarnytskyy (Lviv)

MSC:

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

Keywords:

Siegel modular forms; theta functions; field of modular functions; \(\delta\)-derivatives; Picard-Fuchs extension; Picard-Vessiot extensions

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