On the transcendence degree of the differential field generated by Siegel modular forms

It is a classical fact that the elliptic modular function λ=(ϑ₁₀/ϑ₀₀)⁴ satisfies an algebraic differential equation of order 3 (this goes back to Jacobi’s Fundamenta nova), and none of lower order (cf. [Ra], [M]). In this paper, we show how these properties generalize to Siegel modular functions of arbitrary degree.