On a dynamical Brauer–Manin obstruction

Liang-Chung Hsia[1]; Joseph Silverman[2]

  • [1] Department of Mathematics National Central University Chung-Li, 32054 Taiwan, R. O. C.
  • [2] Mathematics Department Box 1917 Brown University Providence, RI 02912 USA

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 1, page 235-250
  • ISSN: 1246-7405

Abstract

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Let ϕ : X X be a morphism of a variety defined over a number field  K , let  V X be a K -subvariety, and let  𝒪 ϕ ( P ) = { ϕ n ( P ) : n 0 } be the orbit of a point  P X ( K ) . We describe a local-global principle for the intersection  V 𝒪 ϕ ( P ) . This principle may be viewed as a dynamical analog of the Brauer–Manin obstruction. We show that the rational points of  V ( K ) are Brauer–Manin unobstructed for power maps on  2 in two cases: (1)  V is a translate of a torus. (2)  V is a line and  P has a preperiodic coordinate. A key tool in the proofs is the classical Bang–Zsigmondy theorem on primitive divisors in sequences. We also prove analogous local-global results for dynamical systems associated to endomoprhisms of abelian varieties.

How to cite

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Hsia, Liang-Chung, and Silverman, Joseph. "On a dynamical Brauer–Manin obstruction." Journal de Théorie des Nombres de Bordeaux 21.1 (2009): 235-250. <http://eudml.org/doc/10874>.

@article{Hsia2009,
abstract = {Let $\varphi :X\rightarrow X$ be a morphism of a variety defined over a number field $K$, let $V\subset X$ be a $K$-subvariety, and let $\{\mathcal\{O\}\}_\varphi (P)=\lbrace \varphi ^n(P):n\ge 0\rbrace $ be the orbit of a point $P\in X(K)$. We describe a local-global principle for the intersection $V\cap \{\mathcal\{O\}\}_\varphi (P)$. This principle may be viewed as a dynamical analog of the Brauer–Manin obstruction. We show that the rational points of $V(K)$ are Brauer–Manin unobstructed for power maps on $\mathbb\{P\}^2$ in two cases: (1) $V$ is a translate of a torus. (2) $V$ is a line and $P$ has a preperiodic coordinate. A key tool in the proofs is the classical Bang–Zsigmondy theorem on primitive divisors in sequences. We also prove analogous local-global results for dynamical systems associated to endomoprhisms of abelian varieties.},
affiliation = {Department of Mathematics National Central University Chung-Li, 32054 Taiwan, R. O. C.; Mathematics Department Box 1917 Brown University Providence, RI 02912 USA},
author = {Hsia, Liang-Chung, Silverman, Joseph},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {arithmetic dynamical systems; local-global principle; Brauer–Manin obstruction; Arithmetic dynamics; Brauer-Manin obstruction},
language = {eng},
number = {1},
pages = {235-250},
publisher = {Université Bordeaux 1},
title = {On a dynamical Brauer–Manin obstruction},
url = {http://eudml.org/doc/10874},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Hsia, Liang-Chung
AU - Silverman, Joseph
TI - On a dynamical Brauer–Manin obstruction
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 1
SP - 235
EP - 250
AB - Let $\varphi :X\rightarrow X$ be a morphism of a variety defined over a number field $K$, let $V\subset X$ be a $K$-subvariety, and let ${\mathcal{O}}_\varphi (P)=\lbrace \varphi ^n(P):n\ge 0\rbrace $ be the orbit of a point $P\in X(K)$. We describe a local-global principle for the intersection $V\cap {\mathcal{O}}_\varphi (P)$. This principle may be viewed as a dynamical analog of the Brauer–Manin obstruction. We show that the rational points of $V(K)$ are Brauer–Manin unobstructed for power maps on $\mathbb{P}^2$ in two cases: (1) $V$ is a translate of a torus. (2) $V$ is a line and $P$ has a preperiodic coordinate. A key tool in the proofs is the classical Bang–Zsigmondy theorem on primitive divisors in sequences. We also prove analogous local-global results for dynamical systems associated to endomoprhisms of abelian varieties.
LA - eng
KW - arithmetic dynamical systems; local-global principle; Brauer–Manin obstruction; Arithmetic dynamics; Brauer-Manin obstruction
UR - http://eudml.org/doc/10874
ER -

References

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  1. A. S. Bang, Taltheoretiske Undersogelser. Tidsskrift Mat. 4(5) (1886), 70–80, 130–137. 
  2. G. D. Birkhoff and H. S. Vandiver, On the integral divisors of a n - b n . Ann. of Math. (2) 5(4) (1904), 173–180. Zbl35.0205.01MR1503541
  3. J. Cheon and S. Hahn, The orders of the reductions of a point in the Mordell-Weil group of an elliptic curve. Acta Arith. 88(3) (1999), 219–222. Zbl0933.11029MR1683630
  4. B. Poonen and J. F. Voloch, The Brauer–Manin obstruction for subvarieties of abelian varieties over function fields. Annals of Math., to appear. Zbl1294.11110
  5. L. P. Postnikova and A. Schinzel, Primitive divisors of the expression a n - b n in algebraic number fields. Mat. Sb. (N.S.) 75(117) (1968), 171–177. Zbl0195.33704MR223330
  6. V. Scharaschkin, Local-global problems and the Brauer–Manin obstruction. PhD thesis, University of Michigan, 1999. Zbl0938.11053
  7. A. Schinzel, Primitive divisors of the expression A n - B n in algebraic number fields. J. Reine Angew. Math. 268/269 (1974), 27–33. Zbl0287.12014MR344221
  8. J.-P. Serre, Sur les groupes de congruence des variétés abéliennes. II. Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 731–737. Zbl0222.14025MR289513
  9. J. H. Silverman, Wieferich’s criterion and the a b c -conjecture. J. Number Theory 30(2) (1988), 226–237. Zbl0654.10019MR961918
  10. J. H. Silverman, The Arithmetic of Elliptic Curves. Volume 106 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1986. Zbl0585.14026MR817210
  11. M. Stoll, Finite descent obstructions and rational points on curves. Algebra & Number Theory 1 (2007), 349–391. Zbl1167.11024MR2368954
  12. S.-W. Zhang, Distributions in algebraic dynamics. In Differential Geometry: A Tribute to Professor S.-S. Chern, Surv. Differ. Geom., Vol. X, pages 381–430. Int. Press, Boston, MA, 2006. Zbl1207.37057MR2408228
  13. K. Zsigmondy, Zur Theorie der Potenzreste. Monatsh. Math. Phys. 3(1) (1892), 265–284. MR1546236

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