Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-06-22T16:43:55.723Z Has data issue: false hasContentIssue false
  • English
  • Français

Local-global principles for Weil–Châtelet divisibility in positive characteristic

Published online by Cambridge University Press:  01 February 2017

BRENDAN CREUTZ  and
JOSÉ FELIPE VOLOCH
BRENDAN CREUTZ
Affiliation:
School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand. e-mail: brendan.creutz@canterbury.ac.nz
JOSÉ FELIPE VOLOCH
Affiliation:
School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand and Department of Mathematics, University of Texas, Austin, TX 78712, U.S.A. e-mail: felipe.voloch@canterbury.ac.nz
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend existing results characterizing Weil-Châtelet divisibility of locally trivial torsors over number fields to global fields of positive characteristic. Building on work of González-Avilés and Tan, we characterize when local-global divisibility holds in such contexts, providing examples showing that these results are optimal. We give an example of an elliptic curve over a global field of characteristic 2 containing a rational point which is locally divisible by 8, but is not divisible by 8 as well as examples showing that the analogous local-global principle for divisibility in the Weil-Châtelet group can also fail.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 2 , September 2017 , pp. 357 - 367
Copyright
Copyright © Cambridge Philosophical Society 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.)

References

REFERENCES

[Baš72] Bašmakov, M. I. Cohomology of Abelian varieties over a number field. Uspehi Mat. Nauk 27 (1972). no. 6(168), 25–66 (Russian).Google Scholar
[Cas62a] Cassels, J. W. S. Arithmetic on curves of genus 1. III. The Tate-Šafarevič and Selmer groups. Proc. London Math. Soc. (3) 12 (1962), 259–296.CrossRefGoogle Scholar
[Cas62b] Cassels, J. W. S. Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung. J. Reine Angew. Math. 211, (1962), 95–112.CrossRefGoogle Scholar
[CF96] Cassels, J. W. S. and Flynn, E. V. Prolegomena to a middlebrow arithmetic of curves of genus 2. London Math. Soc. Lecture Note Series, vol 230 (Cambridge University Press, Cambridge, 1996).Google Scholar
[ÇS15] Çiperiani, M. and Stix, J. Weil–Châtelet divisible elements in Tate-Shafarevich groups II: On a question of Cassels. J. Reine Angew. Math. 700 (2015), 175–207.CrossRefGoogle Scholar
[CGP15] Conrad, B., Gabber, O. and Prasad, G. Pseudo-reductive groups, 2nd ed. New Mathematical Monogr. vol. 26 (Cambridge University Press, Cambridge, 2015).CrossRefGoogle Scholar
[Cre13] Creutz, B. Locally trivial torsors that are not Weil–Châtelet divisible. Bull. Lond. Math. Soc. 45 (2013), no. 5. 935–942.CrossRefGoogle Scholar
[Cre16] Creutz, B. On the local-global principle for divisibility in the cohomology of elliptic curves. Math. Res. Lett. 23 (2016), no. 2, 377–387.CrossRefGoogle Scholar
[DZ01] Dvornicich, R. and Zannier, U. Local-global divisibility of rational points in some commutative algebraic groups. Bull. Soc. Math. France 129 (2001), no. 3, 317–338. (English, with English and French summaries).CrossRefGoogle Scholar
[DZ04] Dvornicich, R. and Zannier, U. An analogue for elliptic curves of the Grunwald–Wang example. C. R. Math. Acad. Sci. Paris 338 (2004), no. 1, 47–50. (English, with English and French summaries).CrossRefGoogle Scholar
[DZ07] Dvornicich, R. and Zannier, U. On a local-global principle for the divisibility of a rational point by a positive integer. Bull. Lond. Math. Soc. 39 (2007), no. 1, 27–34.CrossRefGoogle Scholar
[FZ86] Fried, M. D. and Jarden, M. Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] vol 11 (Springer-Verlag, Berlin, 1986).CrossRefGoogle Scholar
[GS06] Gille, P. and Szamuely, T. Central simple algebras and Galois cohomology. Camb. Stud. Adv. Math. vol. 101 (Cambridge University Press, Cambridge, 2006).Google Scholar
[GA09] González-Avilés, C. D. Arithmetic duality theorems for 1-motives over function fields. J. Reine Angew. Math. 632 (2009), 203–231.Google Scholar
[GAT12] González-Avilés, C. D. and Tan, K.-S. On the Hasse principle for finite group schemes over global function fields. Math. Res. Lett. 19 (2012), no. 2, 453–460.CrossRefGoogle Scholar
[Ill79] Illusie, L. Complexe de de Rham-Witt et cohomologie cristalline. Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 4, 501–661.CrossRefGoogle Scholar
[KM85] Katz, N. M. and Mazur, B. Arithmetic moduli of elliptic curves. Annals of Math. Studies, vol. 108 (Princeton University Press, Princeton, NJ, 1985).Google Scholar
[LW] Lawson, T. and Wuthrich, C. Vanishing of some cohomology groups for elliptic curves. available at arXiv:1505.02940v2.Google Scholar
[Mil86] Milne, J. S. Arithmetic duality theorems. Perspectives in Mathematics. vol. 1 (Academic Press, Inc., Boston, MA, 1986).Google Scholar
[Mil75] Milne, J. S. On a conjecture of Artin and Tate . Annals of Math. Second Series 102 (1975), no. 3, 517–533.CrossRefGoogle Scholar
[NSW08] Neukirch, J., Schmidt, Al. and Wingberg, K. Cohomology of number fields. 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 323 (Springer-Verlag, Berlin, 2008).CrossRefGoogle Scholar
[PRV12] Paladino, L., Ranieri, G. and Viada, E. On local-global divisibility by pn in elliptic curves. Bull. Lond. Math. Soc. 44 (2012), no. 4, 789–802.CrossRefGoogle Scholar
[PRV14] Paladino, L., Ranieri, G. and Viada, E. On the minimal set for counterexamples to the local-global principle. J. Algebra 415 (2014), 290–304.CrossRefGoogle Scholar
[PS99] Poonen, B. and Stoll, M. The Cassels–Tate pairing on polarized abelian varieties. Ann. of Math. (2) 150 (1999), no. 3, 1109–1149.CrossRefGoogle Scholar
[Ser64] Serre, J.-P. Sur les groupes de congruence des variétés abéliennes. Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 3–20. (French, with Russian summary).Google Scholar
[Tat64] Tate, J. On the conjectures of Birch and Swinnerton–Dyer and a geometric analog (1964). 415–440. Séminaire Bourbaki. Vol. 9, Exp. No. 306.Google Scholar
[Tat63] Tate, J. Duality theorems in Galois cohomology over number fields. Proc. Internat. Congr. Math. (Stockholm, 1962) (Inst. Mittag–Leffler, Djursholm, 1963), pp. 288–295.Google Scholar
[Ulm91] Ulmer, D. L. p-descent in characteristic p . Duke Math. J. 62 (1991), no. 2, 237–265.CrossRefGoogle Scholar
[Vol90] Voloch, J. F. Explicit p-descent for elliptic curves in characteristic p . Compositio Math. 74 (1990), no. 3, 247–258.Google Scholar