Research PapersNo Access

A local–global principle in the dynamics of quadratic polynomials

David Krumm

Department of Mathematics, Colby College, 5847 Mayflower Hill, Waterville, Maine 04901, USA

https://doi.org/10.1142/S1793042116501360Cited by:8 (Source: Crossref)

Abstract

Let $K$ be a number field, $f \in K [x]$ a quadratic polynomial, and $n \in {1, 2, 3}$ . We show that if $f$ has a point of period $n$ in every non-archimedean completion of $K$ , then $f$ has a point of period $n$ in $K$ . For $n \in {4, 5}$ we show that there exist at most finitely many linear conjugacy classes of quadratic polynomials over $K$ for which this local–global principle fails. By considering a stronger form of this principle, we strengthen global results obtained by Morton and Flynn–Poonen–Schaefer in the case $K = Q$ . More precisely, we show that for every quadratic polynomial $f \in Q [x]$ there exist infinitely many primes $p$ such that $f$ does not have a point of period four in the $p$ -adic field $Q_{p}$ . Conditional on knowing all rational points on a particular curve of genus 11, the same result is proved for points of period five.

Keywords:

AMSC: 11S20, 20B35, 11G30

References

1. D. Berend and Yu. Bilu, Polynomials with roots modulo every integer, Proc. Amer. Math. Soc. 124(6) (1996) 1663–1671. Crossref, Web of Science, Google Scholar
2. W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24(3–4) (1997) 235–265; Computational algebra and number theory (London, 1993). Crossref, Web of Science, Google Scholar
3. T. Bousch, Sur quelques problèmes de dynamique holomorphe, Ph.D. thesis, Université de Paris-Sud, Centre d'Orsay (1992). Google Scholar
4. J. J. Cannon, B. C. Cox and D. F. Holt, Computing the subgroups of a permutation group, J. Symbolic Comput. 31(1–2) (2001) 149–161; Special Issue on Computational Algebra and Number Theory (Milwaukee, WI, 1996). Crossref, Web of Science, Google Scholar
5. D. A. Cox, Primes of the Form x2 + ny2, 2nd edn., Pure and Applied Mathematics (Hoboken) (Wiley, Hoboken, NJ, 2013). Crossref, Google Scholar
6. J. R. Doyle, X. Faber and D. Krumm, Preperiodic points for quadratic polynomials over quadratic fields, New York J. Math. 20 (2014) 507–605. Web of Science, Google Scholar
7. J. R. Doyle, D. Krumm and J. L. Wetherell, Classification of preperiodic structures for quadratic polynomials over quadratic fields, in preparation. Google Scholar
8. E. V. Flynn, B. Poonen and E. F. Schaefer, Cycles of quadratic polynomials and rational points on a genus-2 curve, Duke Math. J. 90(3) (1997) 435–463. Crossref, Web of Science, Google Scholar
9. G. J. Janusz, Algebraic Number Fields, 2nd edn., Graduate Studies in Mathematics, Vol. 7 (American Mathematical Society, Providence, RI, 1996). Google Scholar
10. D. Krumm, Code for the main computations in the article “a local–global principle in the dynamics of quadratic polynomials”, preprint (2015); https://github.com/davidkrumm/local-global. Google Scholar
11. J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem, in Algebraic Number Fields: L-Functions and Galois Properties (Academic Press, London, 1977), pp. 409–464. Google Scholar
12. J. S. Leon, Partitions, refinements, and permutation group computation, in Groups and Computation, II, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 28 (American Mathematical Society, Providence, RI, 1997), pp. 123–158. Crossref, Google Scholar
13. P. Morton, On certain algebraic curves related to polynomial maps, Compos. Math. 103(3) (1996) 319–350. Web of Science, Google Scholar
14. P. Morton, Arithmetic properties of periodic points of quadratic maps, II, Acta Arith. 87(2) (1998) 89–102. Crossref, Web of Science, Google Scholar
15. P. Morton and P. Patel, The Galois theory of periodic points of polynomial maps, Proc. London Math. Soc. (3) 68(2) (1994) 225–263. Crossref, Google Scholar
16. J. Neukirch, Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 322 (Springer, Berlin, 1999). Crossref, Google Scholar
17. C. Panraksa, Arithmetic dynamics of quadratic polynomials and dynamical units, Ph.D. thesis, University of Maryland, College Park (2011). Google Scholar
18. B. Poonen, The classification of rational preperiodic points of quadratic polynomials over $Q$ : A refined conjecture, Math. Z. 228(1) (1998) 11–29. Crossref, Web of Science, Google Scholar
19. J.-P. Serre, Lectures on the Mordell–Weil Theorem, 3rd edn., Aspects of Mathematics (Friedr. Vieweg & Sohn, Braunschweig, 1997). Crossref, Google Scholar
20. J.-P. Serre, Topics in Galois Theory, 2nd edn., Research Notes in Mathematics, Vol. 1 (A. K. Peters, Ltd., Wellesley, MA, 2008). Google Scholar
21. J. H. Silverman, The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics, Vol. 241 (Springer, New York, 2007). Crossref, Google Scholar
22. W. A. Stein et al., Sage Mathematics Software, Version 6.9 (The Sage Development Team, 2015); http://www.sagemath.org. Google Scholar