  It is a classical fact that the elliptic modular functions satisfies an
algebraic differential equation of order 3, and none of lower order. We show
how this generalizes to Siegel modular functions of arbitrary degree. The key
idea is that the partial differential equations they satisfy are governed by
Gauss--Manin connections, whose monodromy groups are well-known. Modular theta
functions provide a concrete interpretation of our result, and we study their
differential properties in detail in the case of degree 2.

Number Theory;Algebraic Geometry