Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

The Vojta conjecture implies Galois rigidity in dynamical families


Author: Wade Hindes
Journal: Proc. Amer. Math. Soc. 144 (2016), 1931-1938
MSC (2010): Primary 14G05, 37P55; Secondary 11R32
DOI: https://doi.org/10.1090/proc/12877
Published electronically: December 15, 2015
MathSciNet review: 3460156
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the Vojta (or Hall-Lang) conjecture implies that the arboreal Galois representations in a 1-parameter family of quadratic polynomials are surjective if and only if they surject onto some finite and uniform quotient. As an application, we use the Vojta conjecture, our uniformity theorem over $ \mathbb{Q}(t)$, and Hilbert's irreducibility theorem to prove that the prime divisors of many quadratic orbits have density zero.


References [Enhancements On Off] (What's this?)

  • [1] Nigel Boston and Rafe Jones, The image of an arboreal Galois representation, Pure Appl. Math. Q. 5 (2009), no. 1, 213-225. MR 2520459 (2010e:11050), https://doi.org/10.4310/PAMQ.2009.v5.n1.a6
  • [2] C. Gratton, K. Nguyen, and T. J. Tucker, $ ABC$ implies primitive prime divisors in arithmetic dynamics, Bull. Lond. Math. Soc. 45 (2013), no. 6, 1194-1208. MR 3138487, https://doi.org/10.1112/blms/bdt049
  • [3] Wade Hindes, Galois uniformity in quadratic dynamics over $ k(t)$, J. Number Theory 148 (2015), 372-383. MR 3283185, https://doi.org/10.1016/j.jnt.2014.09.033
  • [4] W. Hindes, The arithmetic of curves defined by iteration. Acta Arith., in press, arXiv:1305.0222, (2014).
  • [5] Su-Ion Ih, Height uniformity for algebraic points on curves, Compositio Math. 134 (2002), no. 1, 35-57. MR 1931961 (2004g:11052), https://doi.org/10.1023/A:1020246809487
  • [6] Patrick Ingram, Lower bounds on the canonical height associated to the morphism $ \phi (z)=z^d+c$, Monatsh. Math. 157 (2009), no. 1, 69-89. MR 2504779 (2010a:11122), https://doi.org/10.1007/s00605-008-0018-6
  • [7] Rafe Jones, An iterative construction of irreducible polynomials reducible modulo every prime, J. Algebra 369 (2012), 114-128. MR 2959789, https://doi.org/10.1016/j.jalgebra.2012.05.020
  • [8] Rafe Jones, Galois representations from pre-image trees: an arboreal survey, Actes de la Conférence ``Théorie des Nombres et Applications'', Publ. Math. Besançon Algèbre Théorie Nr., Presses Univ. Franche-Comté, Besançon, 2013, pp. 107-136 (English, with English and French summaries). MR 3220023
  • [9] Rafe Jones, The density of prime divisors in the arithmetic dynamics of quadratic polynomials, J. Lond. Math. Soc. (2) 78 (2008), no. 2, 523-544. MR 2439638 (2010b:37239), https://doi.org/10.1112/jlms/jdn034
  • [10] Michal Křížek, Florian Luca, and Lawrence Somer, 17 lectures on Fermat numbers. From number theory to geometry; With a foreword by Alena Šolcová, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 9, Springer-Verlag, New York, 2001. MR 1866957 (2002i:11001)
  • [11] Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers, 3rd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. MR 2078267 (2005c:11131)
  • [12] Jean-Pierre Serre, Topics in Galois theory, Research Notes in Mathematics. Lecture notes prepared by Henri Damon [Henri Darmon]; With a foreword by Darmon and the author, vol. 1, Jones and Bartlett Publishers, Boston, MA, 1992. MR 1162313 (94d:12006)
  • [13] Joseph H. Silverman, Primitive divisors, dynamical Zsigmondy sets, and Vojta's conjecture, J. Number Theory 133 (2013), no. 9, 2948-2963. MR 3057058, https://doi.org/10.1016/j.jnt.2013.03.005
  • [14] Joseph H. Silverman, The arithmetic of dynamical systems, Graduate Texts in Mathematics, vol. 241, Springer, New York, 2007. MR 2316407 (2008c:11002)
  • [15] Joseph H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094 (2010i:11005)
  • [16] Michael Stoll, Rational points on curves, J. Théor. Nombres Bordeaux 23 (2011), no. 1, 257-277 (English, with English and French summaries). MR 2780629 (2012d:14037)
  • [17] Adrian Vasiu, Surjectivity criteria for $ p$-adic representations. I, Manuscripta Math. 112 (2003), no. 3, 325-355. MR 2067042 (2005e:11157), https://doi.org/10.1007/s00229-003-0402-4

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14G05, 37P55, 11R32

Retrieve articles in all journals with MSC (2010): 14G05, 37P55, 11R32


Additional Information

Wade Hindes
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912

DOI: https://doi.org/10.1090/proc/12877
Keywords: Rational points on curves, arithmetic dynamics, Galois theory
Received by editor(s): January 7, 2015
Received by editor(s) in revised form: June 16, 2015
Published electronically: December 15, 2015
Communicated by: Romyar T. Sharifi
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society