Geodesic
Derivatives of Siegel modular forms and exponential functions
Izvestiya. Mathematics , Tome 65 (2001) no. 4, pp. 659-672.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{IM2_2001_65_4_a1,
     author = {D. Bertrand and W. V. Zudilin},
     title = {Derivatives of {Siegel} modular forms and exponential functions},
     journal = {Izvestiya. Mathematics },
     pages = {659--672},
     publisher = {mathdoc},
     volume = {65},
     number = {4},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2001_65_4_a1/}
}
TY  - JOUR
AU  - D. Bertrand
AU  - W. V. Zudilin
TI  - Derivatives of Siegel modular forms and exponential functions
JO  - Izvestiya. Mathematics 
PY  - 2001
SP  - 659
EP  - 672
VL  - 65
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2001_65_4_a1/
LA  - en
ID  - IM2_2001_65_4_a1
ER  - 
%0 Journal Article
%A D. Bertrand
%A W. V. Zudilin
%T Derivatives of Siegel modular forms and exponential functions
%J Izvestiya. Mathematics 
%D 2001
%P 659-672
%V 65
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2001_65_4_a1/
%G en
%F IM2_2001_65_4_a1
D. Bertrand; W. V. Zudilin. Derivatives of Siegel modular forms and exponential functions. Izvestiya. Mathematics , Tome 65 (2001) no. 4, pp. 659-672. http://geodesic.mathdoc.fr/item/IM2_2001_65_4_a1/

[1] Mahler K., “On algebraic differential equations satisfied by automorphic functions”, J. Austral. Math. Soc., 10 (1969), 445–450 | DOI | MR | Zbl

[2] Nishioka K., “A conjecture of Mahler on automorphic functions”, Arch. Math. (Basel), 53:1 (1989), 46–51 | MR | Zbl

[3] Bertrand D., Zudilin W., “On the transcendence degree of the differential field generated by Siegel modular forms”, Prépubl. de l'Institut de Math. de Jussieu, 248, 2000 ; arXiv: math/0006176 | Zbl

[4] Zudilin W., Number theory casting a look at the mirror, Preprint, 2000, submitted for publication ; arXiv: math/0008237 | Zbl

[5] Van der Varden B. L., Algebra, Nauka, M., 1976 | MR

[6] Ax J., “On Schanuel's conjectures”, Ann. of Math. (2), 93 (1971), 252–268 | DOI | MR | Zbl

[7] Kolchin E. R., Differential algebra and algebraic groups, Pure Appl. Math., 54, Academic Press, N. Y.–London, 1973 | MR | Zbl

[8] Uitteker E. T., Vatson Dzh. N., Kurs sovremennogo analiza, T. 2, Fizmatlit, M., 1963

[9] Kontsevich M., “Product formulas for modular forms on $O(2,n)$ (after R. Borcherds)”, [Exp. no. 821], Astérisque, 245 (1997), 41–56 ; Sém. Bourbaki, Exp. 820–834, 1996/97 ; arXiv: alg-geom/9709006 | MR | Zbl | MR

[10] Gritsenko V. A., Nikulin V. V., “Automorphic forms and Lorentzian Kac–Moody algebras, II”, Internat. J. Math., 9:2 (1998), 201–275 | DOI | MR | Zbl