An abelian analogue of Schanuel’s conjecture and applications

  • Patrice Philippon
  • Biswajyoti SahaEmail author
  • Ekata Saha


In this article we study an abelian analogue of Schanuel’s conjecture. This conjecture falls in the realm of the generalised period conjecture of André. As shown by Bertolin, the generalised period conjecture includes Schanuel’s conjecture as a special case. Extending methods of Bertolin, it can be shown that the abelian analogue of Schanuel’s conjecture we consider also follows from André’s conjecture. Cheng et al. showed that the classical Schanuel’s conjecture implies the algebraic independence of the values of the iterated exponential function and the values of the iterated logarithmic function, answering a question of Waldschmidt. We then investigate a similar question in the setup of abelian varieties.


Abelian analogue of Schanuel’s conjecture Linear disjointness Algebraic independence 

Mathematics Subject Classification

11J81 11J89 11J95 



The authors would like to thank D. Bertrand for helpful discussions and useful comments on an earlier version of this article. The second and the third authors would like to thank the Institut de Mathématiques de Jussieu for hospitality during academic visits in the frame of the IRSES Moduli and LIA. The authors would also like to thank the referee for helpful suggestions.


  1. 1.
    André, Y.: Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part. Compos. Math. 82(1), 1–24 (1992)MathSciNetzbMATHGoogle Scholar
  2. 2.
    André, Y.: Une introduction aux motifs, Panoramas et Synthèses 17. Société Math. France, Paris (2004)Google Scholar
  3. 3.
    Bertolin, C.: Périodes de 1-motifs et transcendance. J. Number Theory 97(2), 204–221 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Chandrasekharan, K.: Elliptic Functions, Grundlehren der Mathematischen Wissenschaften 281. Springer, Berlin (1985)Google Scholar
  5. 5.
    Cheng, C., Dietel, B., Herblot, M., Huang, J., Krieger, H., Marques, D., Mason, J., Mereb, M., Wilson, S.R.: Some consequences of Schanuel’s conjecture. J. Number Theory 129(6), 1464–1467 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Deligne, P.: Hodge Cycles on Abelian Varieties, Hodge Cycles, Motives and Shimura Varieties, Lecture Notes in Math, pp. 6–77. Springer, New York (1975)Google Scholar
  7. 7.
    Lang, S.: Algebra, Graduate Texts in Math, 3rd edn. Springer, New York (2002)Google Scholar
  8. 8.
    Philippon, P.: Variétés abéliennes et indépendance algébrique II: Un analogue abélien du théorème de Lindemann–Weierstraß. Invent. Math. 72(3), 389–405 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Vallée, G.: Sur la conjecture de Schanuel abélienne, preprint (13 p.), October 2018Google Scholar
  10. 10.
    Wüstholz, G.: Über das Abelsche Analogon des Lindemannschen Satzes I. Invent. Math. 72(3), 363–388 (1983)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Patrice Philippon
    • 1
  • Biswajyoti Saha
    • 2
    Email author
  • Ekata Saha
    • 3
  1. 1.Équipe de Théorie des Nombres, Institut de Mathématiques de Jussieu-Paris Rive GaucheUMR CNRS 7586ParisFrance
  2. 2.School of Mathematics and StatisticsUniversity of HyderabadHyderabadIndia
  3. 3.Statistics and Mathematics UnitIndian Statistical InstituteKolkataIndia

Personalised recommendations