Glasnik Matematicki, Vol. 51, No. 2 (2016), 335-343.

SOME APPLICATIONS OF THE P-ADIC ANALYTIC SUBGROUP THEOREM

Clemens Fuchs and Duc Hiep Pham

Department of Mathematics, University of Salzburg, Hellbrunnerstr. 34, 5020 Salzburg, Austria
e-mail: clemens.fuchs@sbg.ac.at

University of Education, Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam
e-mail: phamduchiepk6@gmail.com


Abstract.   We use a p-adic analogue of the analytic subgroup theorem of Wüstholz to deduce the transcendence and linear independence of some new classes of p-adic numbers. In particular we give p-adic analogues of results of Wüstholz contained in [20] and generalizations of results obtained by Bertrand in [3, 4].

2010 Mathematics Subject Classification.   11G99, 14L10, 11J86.

Key words and phrases.   Commutative algebraic groups, transcendence theory, p-adic numbers.


Full text (PDF) (access from subscribing institutions only)

DOI: 10.3336/gm.51.2.04


References:

  1. W. Adams, Transcendental numbers in the p-adic domain, Amer. J. Math. 88 (1966), 279-308.
    MathSciNet     CrossRef

  2. A. Baker and G. Wüstholz, Logarithmic forms and Diophantine geometry, Cambridge University Press, Cambridge, 2007.
    MathSciNet    

  3. D. Bertrand, Sous-groupes à un paramètre p-adique de variétés de groupe, Invent. Math. 40 (1977), 171-193.
    MathSciNet     CrossRef

  4. D. Bertrand, Problèmes locaux, Astérisque 69-70 (1979), 163-189.

  5. D. Bertrand, Lemmes de zèros et nombres transcendants, Séminaire Bourbaki Vol. 1985/86, Astérisque 145-146 (1987), 21-44.
    MathSciNet    

  6. N. Bourbaki, Elements of Mathematics. Lie groups and Lie algebras. Part I: Chapters 1-3. English translantion., Actualities scientifiques et industrielles, Herman. Adiwes International Series in Mathematics. Paris: Hermann, Publishers in Arts and Science; Reading, Mass.: Addison-Wesley Publishing Company. XVII, 1975. %??

  7. A. Brumer, On the units of algebraic numbers fields, Mathematika 14 (1967), 121-124.
    MathSciNet     CrossRef

  8. Y. Z. Flicker, Transcendence theory over local fields, PhD dissertation, University of Cambridge, 1978.

  9. Y. Z. Flicker, Linear forms on arithmetic Abelian varieties: ineffective bounds, Mém. Soc. Math. France (N.S.) (1980/81), 41-47.
    MathSciNet    

  10. C. Fuchs and D. H. Pham, Commutative algebraic groups and p-adic linear forms, Acta Arith. 169 (2015), 115-147.
    MathSciNet     CrossRef

  11. C. Fuchs and D. H. Pham, The p-adic analytic subgroup theorem revisited, p-Adic Numbers, Ultrametric Anal. Appl. 7 (2015), 143-156.
    MathSciNet     CrossRef

  12. A. Günther, Über transzendente p-adische Zahlen. I, J. Reine Angew. Math. 192 (1953), 155-166.
    MathSciNet     CrossRef

  13. E. Lutz, Sur l'équation Y2=AX3-AX-B dans les corps p-adiques, J. Reine Angew. Math. 177 (1937), 238-247.
    MathSciNet     CrossRef

  14. K. Mahler, Ein Beweis der Transzendenz der P-adischen Exponentialfunktion, J. Reine Angew. Math. 169 (1933), 61-66.
    MathSciNet     CrossRef

  15. K. Mahler, Uber transzendente P-adische Zahlen, Compositio Math. 2 (1935), 259-275.
    MathSciNet     CrossRef

  16. T. Matev, The p-adic analytic subgroup theorem and applications, http://arxiv.org/pdf/1010.3156v1.pdf.

  17. A. M. Robert, A course in p-adic analysis, GTM 198, Springer-Verlag, 2000.
    MathSciNet     CrossRef

  18. G. Veldkamp, Ein Transzendenz-Satz für p-adische Zahlen, J. London Math. Soc. 15 (1940), 183-192.
    MathSciNet     CrossRef

  19. A. Weil, Sur les fonctions elliptiques p-adiques, C. R. Acad. Sc. Paris 203 (1936), 22-24.

  20. G. Wüstholz, Some remarks on a conjecture of Waldschmidt, in: Diophantine approximations and transcendental numbers, Birkhäuser Boston, Boston, MA, 1983, 329-336.
    MathSciNet    

  21. G. Wüstholz, Algebraische Punkte auf Analytischen Untergruppen algebraischer Gruppen, Ann. of Math. (2) 129 (1989), 501-517.
    MathSciNet     CrossRef

  22. G. Wüstholz, Multiplicity estimates on group varieties, Ann. of Math. (2) 129 (1989), 471-500.
    MathSciNet     CrossRef

Glasnik Matematicki Home Page