Hostname: page-component-77f85d65b8-2tv5m Total loading time: 0 Render date: 2026-04-17T22:31:11.979Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on endomorphisms and rational points

Part of: Arithmetic and non-Archimedean dynamical systems Arithmetic problems. Diophantine geometry Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  24 August 2011

E. Amerik ,
F. Bogomolov  and
M. Rovinsky
E. Amerik
Affiliation:
Laboratoire de Mathématiques, Campus Scientifique d’Orsay, Université Paris-Sud, Bâtiment 425, 91405 Orsay, France (email: Ekaterina.Amerik@math.u-psud.fr)
F. Bogomolov
Affiliation:
Courant Institute of Mathematical Sciences, 251 Mercer Str., New York, NY 10012, USA Laboratory of Algebraic Geometry, GU-HSE, 7 Vavilova Str., Moscow 117312, Russia (email: bogomolo@cims.nyu.edu)
M. Rovinsky
Affiliation:
Independent University of Moscow, B. Vlasievsky Per. 11, 119002 Moscow, Russia Institute for Information Transmission Problems of Russian Academy of Sciences(email: marat@mccme.ru)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Let X be an algebraic variety and let f:X−−→X be a rational self-map with a fixed point q, where everything is defined over a number field K. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points on X. As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold, originally proved by Claire Voisin and the first author in 2007.

Keywords

rational pointsrational mapsp-adic neighbourhood

MSC classification

Secondary: 14G05: Rational points 11S82: Non-Archimedean dynamical systems 37P55: Arithmetic dynamics on general algebraic varieties

Information

Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 6 , November 2011 , pp. 1819 - 1842
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[Ame09]Amerik, E., A computation of invariants of a rational self-map, Ann. Fac. Sci. Toulouse Math (6) 18 (2009), 445–457, arXiv:0707.3947.Google Scholar
[AC08]Amerik, E. and Campana, F., Fibrations méromorphes sur certaines variétés à fibré canonique trivial, Pure Appl. Math. Q. 4 (2008), 509–545.Google Scholar
[AV08]Amerik, E. and Voisin, C., Potential density of rational points on the variety of lines of a cubic fourfold, Duke Math. J. 145 (2008), 379–408, arXiv:0707.3948.CrossRefGoogle Scholar
[Arn88]Arnold, V. I., Geometrical methods in the theory of ordinary differential equations, Grundlehren der Mathematischen Wissenschaften, vol. 250, second edition (Springer, New York, 1988).Google Scholar
[AM69]Atiyah, M. F. and Macdonald, I. G., Introduction to commutative algebra (Addison-Wesley, Reading, MA, 1969).Google Scholar
[Bea83]Beauville, A., Some remarks on Kähler manifolds with c 1=0, in Classification of algebraic and analytic manifolds (Katata, 1982), Progress in Mathematics, vol. 39 (Birkhäuser Boston, Boston, MA, 1983), 1–26.Google Scholar
[BD85]Beauville, A. and Donagi, R., La variété des droites d’une hypersurface cubique de dimension 4, C. R. Acad. Sci. Paris Sér. I 301 (1985), 703–706.Google Scholar
[BT00]Bogomolov, F. and Tschinkel, Yu., Density of rational points on elliptic K3 surfaces, Asian J. Math. 4 (2000), 351–368.CrossRefGoogle Scholar
[Cam04]Campana, F., Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble) 54 (2004), 499–630.CrossRefGoogle Scholar
[Che]Chen, X., Self rational maps of K3 surfaces, arXiv:math/1008.1619.Google Scholar
[CG72]Clemens, H. and Griffiths, P., The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356.CrossRefGoogle Scholar
[Eis95]Eisenbud, D., Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150 (Springer, New York, 1995).Google Scholar
[Fak03]Fakhruddin, N., Questions on self maps of algebraic varieties, J. Ramanujan Math. Soc. 18 (2003), 109–122, arXiv:math/0212208.Google Scholar
[GT09]Ghioca, D. and Tucker, T., Periodic points, linearizing maps, and the dynamical Mordell–Lang problem, J. Number Theory 129 (2009), 1392–1403.CrossRefGoogle Scholar
[HT00]Hassett, B. and Tschinkel, Yu., Abelian fibrations and rational points on symmetric products, Internat. J. Math. 11 (2000), 1163–1176.CrossRefGoogle Scholar
[HT]Hassett, B. and Tschinkel, Yu., Flops on holomorphic symplectic fourfolds and determinantal cubic hypersurfaces, arXiv:0805.4162.Google Scholar
[HY83]Herman, M. and Yoccoz, J.-C., Generalizations of some theorems of small divisors to non-Archimedean fields, in Geometric dynamics (Rio de Janeiro, 1981), Lecture Notes in Mathematics, vol. 1007 (Springer, Berlin, 1983), 408–447.CrossRefGoogle Scholar
[Jou78]Jouanolou, J.-P., Hypersurfaces solutions d’une équation de Pfaff analytique, Math. Ann. 232 (1978), 239–245.Google Scholar
[Kob98]Kobayashi, S., Hyperbolic complex spaces (Springer, Berlin, Heidelberg, 1998).CrossRefGoogle Scholar
[NZ07]Nakayama, N. and Zhang, D.-Q., Building blocks of étale endomorphisms of complex projective manifolds, Preprint no. 1577 (Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 2007), http://www.kurims.kyoto-u.ac.jp/preprint/preprint_y2007.html 399.Google Scholar
[Pac03]Pacienza, G., Rational curves on general projective hypersurfaces, J. Algebraic Geom. 12 (2003), 245–267.CrossRefGoogle Scholar
[Poi28]Poincaré, H., Œuvres, 1 (Gauthier-Villars, Paris, 1928), 36–129.Google Scholar
[Ron08]Rong, F., Linearization of holomorphic germs with quasi-parabolic fixed points, Ergod. Th. & Dynam. Sys. 28 (2008), 979–986.CrossRefGoogle Scholar
[Ser92]Serre, J.-P., Topics in Galois theory (Jones and Bartlett, Boston, 1992).Google Scholar
[Ter85]Terasoma, T., Complete intersections with middle Picard number 1 defined over ℚ, Math. Z. 189 (1985), 289–296.CrossRefGoogle Scholar
[vL07]van Luijk, R., K3 surfaces with Picard number one and infinitely many rational points, Algebra Number Theory 1 (2007), 1–15.Google Scholar
[Voi98]Voisin, C., A correction: ‘On a conjecture of Clemens on rational curves on hypersurfaces’, J. Differential Geom. 49 (1998), 601–611.Google Scholar
[Voi04]Voisin, C., Intrinsic pseudo-volume forms and K-correspondences, in The Fano Conference (Università Torino, Turin, 2004), 761–792.Google Scholar
[Yu90]Yu, K., Linear forms in p-adic logarithms. II, Compositio Math. 74 (1990), 15–113.Google Scholar
[Zha06]Zhang, S.-W., Distributions in algebraic dynamics, in Surveys in differential geometry, Vol. X (International Press, Somerville, MA, 2006), 381–430.Google Scholar