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Izv. RAN. Ser. Mat., 2001, Volume 65, Issue 4, Pages 21–34 (Mi izv345)  

This article is cited in 3 scientific papers (total in 3 papers)

Derivatives of Siegel modular forms and exponential functions

D. Bertranda, W. V. Zudilinb

a Université Pierre & Marie Curie, Paris VI
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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.

DOI: https://doi.org/10.4213/im345

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English version:
Izvestiya: Mathematics, 2001, 65:4, 659–672

Bibliographic databases:

MSC: Primary 11F46, 11J81; Secondary 12H05, 14K25, 42A16
Received: 26.12.2000

Citation: D. Bertrand, W. V. Zudilin, “Derivatives of Siegel modular forms and exponential functions”, Izv. RAN. Ser. Mat., 65:4 (2001), 21–34; Izv. Math., 65:4 (2001), 659–672

Citation in format AMSBIB
\Bibitem{BerZud01}
\by D.~Bertrand, W.~V.~Zudilin
\paper Derivatives of Siegel modular forms and exponential functions
\jour Izv. RAN. Ser. Mat.
\yr 2001
\vol 65
\issue 4
\pages 21--34
\mathnet{http://mi.mathnet.ru/izv345}
\crossref{https://doi.org/10.4213/im345}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1857708}
\zmath{https://zbmath.org/?q=an:1021.11013}
\transl
\jour Izv. Math.
\yr 2001
\vol 65
\issue 4
\pages 659--672
\crossref{https://doi.org/10.1070/IM2001v065n04ABEH000345}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33746781039}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Pellarin F., “Les nilradicaux différentiels d'anneaux associés aux groupes triangulaires de Riemann-Schwarz [Differential nilradicals of rings associated with Riemann-Schwarz triangular groups]”, Rend. Sem. Mat. Univ. Padova, 114 (2005), 213–239  mathscinet  zmath  isi
    2. Jonathan Pila, “O-minimality and the André-Oort conjecture for C^n”, Ann. Math, 173:3 (2011), 1779  crossref  mathscinet  zmath  isi  scopus
    3. Habegger Ph., Pila J., “O-minimality and certain atypical intersections”, Ann. Sci. Ec. Norm. Super., 49:4 (2016), 813–858  crossref  mathscinet  zmath  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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