Derivatives of Siegel modular forms and exponential functions. (English. Russian original) Zbl 1021.11013

Bertrand, D.; Zudilin, W.

Izv. Math. 65, No. 4, 659-671 (2001); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 65, No. 4, 21-34 (2001).

Summary: 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 [J. Reine Angew. Math. 554, 47-68 (2003; Zbl 1130.11020)], 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.

MSC:

11F46	Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11J72	Irrationality; linear independence over a field

Keywords:

differential field; Siegel modular forms; exponentials of polynomials; extension of Mahler’s theorem; transcendence degree; derivatives of a modular function

Citations:

Zbl 1130.11020