Derivatives of Siegel modular forms and exponential functions

We show that the differential field generated by Siegel modular forms and the differential field generated by exponentials of polynomials are linearly disjoint over $\mathbb C$. Combined with our previous work [3], this provides a complete multidimensional extension of Mahler's theorem on the transcendence degree of the field generated by the exponential function and the derivatives of a modular function. We give two proofs of our result, one purely algebraic, the other analytic, but both based on arguments from differential algebra and on the stability under the action of the symplectic group of the differential field generated by rational and modular functions.