Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-19T22:19:49.439Z Has data issue: false hasContentIssue false
  • English
  • Français

AX–SCHANUEL CONDITION IN ARBITRARY CHARACTERISTIC

Part of: Differential and difference algebra Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  08 November 2017

Piotr Kowalski
Piotr Kowalski*
Affiliation:
Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland (pkowa@math.uni.wroc.pl)http://www.math.uni.wroc.pl/∼pkowa/
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Ax–Schanuel conjectureformal mapstranscendence

MSC classification

Primary: 11J91: Transcendence theory of other special functions
Secondary: 12H05: Differential algebra
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 18 , Issue 6 , November 2019 , pp. 1157 - 1213
Copyright
© Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Supported by Narodowe Centrum Nauki grants nos. NCN 2012/07/B/ST1/03513, 2015/19/B/ST1/ 01150 and 2015/19/B/ST1/01151; and by ANR Modig (ANR-09-BLAN-0047).

References

Ax, J., On Schanuel’s conjectures, Ann. of Math. 93(2) (1971), 252–268.Google Scholar
Ax, J., Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups, Amer. J. Math. 94 (1972), 1195–1204.Google Scholar
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).Google Scholar
Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron Models, A Series of Modern Surveys in Mathematics Series (Springer, Berlin Heidelberg, 1990).Google Scholar
Brownawell, W. D., Transcendence in positive characteristic, in Number Theory (Tiruchirapalli, 1996), Contemporary Mathematics, Volume 210, pp. 317–332 (American Mathematical Society, Providence, RI, 1998).Google Scholar
Chevalley, C., Une démonstration d’un théorème sur les groupes algébriques, J. Math. Pures Appl. 39 (1960), 307–317.Google Scholar
Conrad, B., A modern proof of Chevalley’s theorem on algebraic groups, J. Ramanujan Math. Soc. 17(1) (2002), 1–18.Google Scholar
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. 327–348 (Birkhäuser, Basel, 2001).Google Scholar
Eisenbud, D., Commutative Algebra with a View Towards Algebraic Geometry (Springer, New York, 1996).Google Scholar
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).Google Scholar
Hartshorne, D., Algebraic Geometry (Springer, New York, 1977).Google Scholar
Hazewinkel, M., Formal Groups and Applications (Academic Press, New York, 1978).Google Scholar
Kirby, J., The theory of the exponential differential equations of semiabelian varieties, Selecta Math. (N.S.) 15(3) (2009), 445–486.Google Scholar
Kowalski, P., Higher differential forms on group schemes. Preprint, in preparation.Google Scholar
Kowalski, P., A note on a theorem of Ax, Ann. Pure Appl. Logic 156 (2008), 96–109.Google Scholar
Kowalski, P., Schanuel property for additive power series, Israel J. Math. 190(1) (2012), 349–363.Google Scholar
Lang, S., Introduction to Transcendental Numbers, Addison-Wesley Series in Mathematics (Addison-Wesley Pub. Co., Reading, Massachusetts, 1966).Google Scholar
Manin, Y. I., The theory of commutative formal groups over fields of finite characteristic, Russian Math. Surveys 18(6) (1963), 1–83.Google Scholar
Matsumura, H., Commutative Algebra, Math. Lecture Notes Series, (Benjamin/Cummings Publishing Company, Reading, Massachusetts, 1980).Google Scholar
Matsumura, H., Commutative Ring Theory (Cambridge University Press, Cambridge, UK, 1986).Google Scholar
Milne, J. S., Étale Cohomology, Princeton Mathematical Series (Princeton University Press, Princeton, New Jersey, 1980).Google Scholar
Moreno, J., Iterative differential Galois theory in positive characteristic: a model theoretic approach, J. Symbolic Logic 76(1) (2011), 125–142.Google Scholar
Pila, J., O-minimality and the André–Oort conjecture for ℂ n , Ann. of Math. (2) 173 (2011), 1779–1840.Google Scholar
Pila, J., Functional transcendence via o-minimality. Available on http://people.maths.ox.ac.uk/pila/LMSNotes.pdf, 2013.Google Scholar
Rosenlicht, M., A note on derivations and differentials on algebraic varieties, Port. Math. 16 (1957), 43–55.Google Scholar
Serre, J.-P., Sur la cohomologie des variétés algébriques, J. Math. Pures Appl. 36(9) (1957).Google Scholar
Serre, J. P., Galois Cohomology, Springer Monographs in Mathematics (Springer, Berlin, Heidelberg, 2002).Google Scholar
Silverman, J. H., The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics (Springer, New York, 1986).Google Scholar
Vojta, P., Jets via Hasse–Schmidt derivations, in Diophantine Geometry, Proceedings (ed. Zannier, U.), Edizioni della Normale, pp. 335–361 (Edizioni della Normale, Pisa, 2006).Google Scholar
Waterhouse, W. C., Introduction to Affine Group Schemes (Springer, New York, 1979).Google Scholar