Skip to main content
×
×
Home

AX–SCHANUEL CONDITION IN ARBITRARY CHARACTERISTIC

Abstract

We prove a positive characteristic version of Ax’s theorem on the intersection of an algebraic subvariety and an analytic subgroup of an algebraic group [Ax, Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups, Amer. J. Math. 94 (1972), 1195–1204]. Our result is stated in a more general context of a formal map between an algebraic variety and an algebraic group. We derive transcendence results of Ax–Schanuel type.

Copyright
References
Hide All
1. Ax J., On Schanuel’s conjectures, Ann. of Math. 93(2) (1971), 252268.
2. Ax J., Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups, Amer. J. Math. 94 (1972), 11951204.
3. Bertrand D., Schanuels conjecture for non-isoconstant elliptic curves over function fields, in Model Theory with Applications to Algebra and Analysis, Vol. 1, (ed. Chatzidakis Z., Macpherson D., Pillay A. and Wilkie A.), LMS Lecture Note Series, Volume 349, (Cambridge University Press, Cambridge, UK, 2008).
4. Bosch S., Lütkebohmert W. and Raynaud M., Néron Models, A Series of Modern Surveys in Mathematics Series (Springer, Berlin Heidelberg, 1990).
5. Brownawell W. D., Transcendence in positive characteristic, in Number Theory (Tiruchirapalli, 1996), Contemporary Mathematics, Volume 210, pp. 317332 (American Mathematical Society, Providence, RI, 1998).
6. Chevalley C., Une démonstration d’un théorème sur les groupes algébriques, J. Math. Pures Appl. 39 (1960), 307317.
7. Conrad B., A modern proof of Chevalley’s theorem on algebraic groups, J. Ramanujan Math. Soc. 17(1) (2002), 118.
8. Denef J. and Loeser F., Geometry on arc spaces of algebraic varieties, in European Congress of Mathematics, Vol. 1 (Barcelona, 2000), Progress in Mathematics, Volume 201, pp. 327348 (Birkhäuser, Basel, 2001).
9. Eisenbud D., Commutative Algebra with a View Towards Algebraic Geometry (Springer, New York, 1996).
10. Grothendieck A., Éléments de géométrie algébrique IV: Étude locale des schémas et des morphismes de schémas. Première partie, Publ. Math. Inst. Hautes Études Sci. 20 (1964).
11. Hartshorne D., Algebraic Geometry (Springer, New York, 1977).
12. Hazewinkel M., Formal Groups and Applications (Academic Press, New York, 1978).
13. Kirby J., The theory of the exponential differential equations of semiabelian varieties, Selecta Math. (N.S.) 15(3) (2009), 445486.
14. Kowalski P., Higher differential forms on group schemes. Preprint, in preparation.
15. Kowalski P., A note on a theorem of Ax, Ann. Pure Appl. Logic 156 (2008), 96109.
16. Kowalski P., Schanuel property for additive power series, Israel J. Math. 190(1) (2012), 349363.
17. Lang S., Introduction to Transcendental Numbers, Addison-Wesley Series in Mathematics (Addison-Wesley Pub. Co., Reading, Massachusetts, 1966).
18. Manin Y. I., The theory of commutative formal groups over fields of finite characteristic, Russian Math. Surveys 18(6) (1963), 183.
19. Matsumura H., Commutative Algebra, Math. Lecture Notes Series, (Benjamin/Cummings Publishing Company, Reading, Massachusetts, 1980).
20. Matsumura H., Commutative Ring Theory (Cambridge University Press, Cambridge, UK, 1986).
21. Milne J. S., Étale Cohomology, Princeton Mathematical Series (Princeton University Press, Princeton, New Jersey, 1980).
22. Moreno J., Iterative differential Galois theory in positive characteristic: a model theoretic approach, J. Symbolic Logic 76(1) (2011), 125142.
23. Pila J., O-minimality and the André–Oort conjecture for ℂ n , Ann. of Math. (2) 173 (2011), 17791840.
24. Pila J., Functional transcendence via o-minimality. Available on http://people.maths.ox.ac.uk/pila/LMSNotes.pdf, 2013.
25. Rosenlicht M., A note on derivations and differentials on algebraic varieties, Port. Math. 16 (1957), 4355.
26. Serre J.-P., Sur la cohomologie des variétés algébriques, J. Math. Pures Appl. 36(9) (1957).
27. Serre J. P., Galois Cohomology, Springer Monographs in Mathematics (Springer, Berlin, Heidelberg, 2002).
28. Silverman J. H., The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics (Springer, New York, 1986).
29. Vojta P., Jets via Hasse–Schmidt derivations, in Diophantine Geometry, Proceedings (ed. Zannier U.), Edizioni della Normale, pp. 335361 (Edizioni della Normale, Pisa, 2006).
30. Waterhouse W. C., Introduction to Affine Group Schemes (Springer, New York, 1979).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 11 *
Loading metrics...

Abstract views

Total abstract views: 42 *
Loading metrics...

* Views captured on Cambridge Core between 8th November 2017 - 24th January 2018. This data will be updated every 24 hours.