The Vojta conjecture implies Galois rigidity in dynamical families
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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 Hilbertâs irreducibility theorem to prove that the prime divisors of many quadratic orbits have density zero.References
- Nigel Boston and Rafe Jones, The image of an arboreal Galois representation, Pure Appl. Math. Q. 5 (2009), no. 1, 213â225. MR 2520459, DOI 10.4310/PAMQ.2009.v5.n1.a6
- C. Gratton, K. Nguyen, and T. J. Tucker, $ABC$ implies primitive prime divisors in arithmetic dynamics, Bull. Lond. Math. Soc. 45 (2013), no. 6, 1194â1208. MR 3138487, DOI 10.1112/blms/bdt049
- Wade Hindes, Galois uniformity in quadratic dynamics over $k(t)$, J. Number Theory 148 (2015), 372â383. MR 3283185, DOI 10.1016/j.jnt.2014.09.033
- W. Hindes, The arithmetic of curves defined by iteration. Acta Arith., in press, arXiv:1305.0222, (2014).
- Su-Ion Ih, Height uniformity for algebraic points on curves, Compositio Math. 134 (2002), no. 1, 35â57. MR 1931961, DOI 10.1023/A:1020246809487
- Patrick Ingram, Lower bounds on the canonical height associated to the morphism $\phi (z)=z^d+c$, Monatsh. Math. 157 (2009), no. 1, 69â89. MR 2504779, DOI 10.1007/s00605-008-0018-6
- Rafe Jones, An iterative construction of irreducible polynomials reducible modulo every prime, J. Algebra 369 (2012), 114â128. MR 2959789, DOI 10.1016/j.jalgebra.2012.05.020
- Rafe Jones, Galois representations from pre-image trees: an arboreal survey, Actes de la ConfĂ©rence âThĂ©orie des Nombres et Applicationsâ, Publ. Math. Besançon AlgĂšbre ThĂ©orie Nr., vol. 2013, Presses Univ. Franche-ComtĂ©, Besançon, 2013, pp. 107â136 (English, with English and French summaries). MR 3220023
- Rafe Jones, The density of prime divisors in the arithmetic dynamics of quadratic polynomials, J. Lond. Math. Soc. (2) 78 (2008), no. 2, 523â544. MR 2439638, DOI 10.1112/jlms/jdn034
- Michal KĆĂĆŸek, Florian Luca, and Lawrence Somer, 17 lectures on Fermat numbers, CMS Books in Mathematics/Ouvrages de MathĂ©matiques de la SMC, vol. 9, Springer-Verlag, New York, 2001. From number theory to geometry; With a foreword by Alena Ć olcovĂĄ. MR 1866957, DOI 10.1007/978-0-387-21850-2
- WĆadysĆaw Narkiewicz, Elementary and analytic theory of algebraic numbers, 3rd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. MR 2078267, DOI 10.1007/978-3-662-07001-7
- Jean-Pierre Serre, Topics in Galois theory, Research Notes in Mathematics, vol. 1, Jones and Bartlett Publishers, Boston, MA, 1992. Lecture notes prepared by Henri Damon [Henri Darmon]; With a foreword by Darmon and the author. MR 1162313
- Joseph H. Silverman, Primitive divisors, dynamical Zsigmondy sets, and Vojtaâs conjecture, J. Number Theory 133 (2013), no. 9, 2948â2963. MR 3057058, DOI 10.1016/j.jnt.2013.03.005
- Joseph H. Silverman, The arithmetic of dynamical systems, Graduate Texts in Mathematics, vol. 241, Springer, New York, 2007. MR 2316407, DOI 10.1007/978-0-387-69904-2
- Joseph H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094, DOI 10.1007/978-0-387-09494-6
- Michael Stoll, Rational points on curves, J. ThĂ©or. Nombres Bordeaux 23 (2011), no. 1, 257â277 (English, with English and French summaries). MR 2780629
- Adrian Vasiu, Surjectivity criteria for $p$-adic representations. I, Manuscripta Math. 112 (2003), no. 3, 325â355. MR 2067042, DOI 10.1007/s00229-003-0402-4
Additional Information
- Wade Hindes
- Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
- MR Author ID: 1022776
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 1931-1938
- MSC (2010): Primary 14G05, 37P55; Secondary 11R32
- DOI: https://doi.org/10.1090/proc/12877
- MathSciNet review: 3460156