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Gonality of dynatomic curves and strong uniform boundedness of preperiodic points

Part of: Curves Arithmetic algebraic geometry Arithmetic and non-Archimedean dynamical systems

Published online by Cambridge University Press:  17 February 2020

John R. Doyle  and
Bjorn Poonen
John R. Doyle
Affiliation:
Department of Mathematics and Statistics, Louisiana Tech University, Ruston,LA 71272, USA email [email protected]
Bjorn Poonen
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge,MA 02139-4307, USA email [email protected]
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Abstract

Fix $d\geqslant 2$ and a field $k$ such that $\operatorname{char}k\nmid d$. Assume that $k$ contains the $d$th roots of $1$. Then the irreducible components of the curves over $k$ parameterizing preperiodic points of polynomials of the form $z^{d}+c$ are geometrically irreducible and have gonality tending to $\infty$. This implies the function field analogue of the strong uniform boundedness conjecture for preperiodic points of $z^{d}+c$. It also has consequences over number fields: it implies strong uniform boundedness for preperiodic points of bounded eventual period, which in turn reduces the full conjecture for preperiodic points to the conjecture for periodic points. Our proofs involve a novel argument specific to finite fields, in addition to more standard tools such as the Castelnuovo–Severi inequality.

Keywords

gonalitydynatomic curvepreperiodic point

MSC classification

Primary: 37P35: Arithmetic properties of periodic points
Secondary: 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields 14H51: Special divisors (gonality, Brill-Noether theory) 37P05: Polynomial and rational maps 37P15: Global ground fields
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 4 , April 2020 , pp. 733 - 743
Copyright
© The Authors 2020

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Footnotes

The second author was supported in part by National Science Foundation grants DMS-1069236 and DMS-1601946 and Simons Foundation grants #402472 (to Bjorn Poonen) and #550033.

References

Bousch, T., Sur quelques problèmes de dynamique holomorphe, PhD thesis, Université de Paris-Sud, Centre d’Orsay (1992).Google Scholar
Doyle, J. R., Krieger, H., Obus, A., Pries, R., Rubinstein-Salzedo, S. and West, L., Reduction of dynatomic curves, Ergodic Theory Dynam. Systems 39 (2019), 2717–2768; MR 4000512.CrossRefGoogle Scholar
Epstein, A., Integrality and rigidity for postcritically finite polynomials, Bull. Lond. Math. Soc. 44 (2012), 39–46; with an appendix by Epstein and Bjorn Poonen; MR 2881322.CrossRefGoogle Scholar
Flynn, E. V., Poonen, B. and Schaefer, E. F., Cycles of quadratic polynomials and rational points on a genus-2 curve, Duke Math. J. 90 (1997), 435–463; MR 1480542 (98j:11048).CrossRefGoogle Scholar
Frey, G., Curves with infinitely many points of fixed degree, Israel J. Math. 85 (1994), 79–83; MR 1264340.CrossRefGoogle Scholar
Gao, Y., Preperiodic dynatomic curves for z↦z d + c, Fund. Math. 233 (2016), 37–69; MR 3460633.Google Scholar
Hutz, B. and Towsley, A., Misiurewicz points for polynomial maps and transversality, New York J. Math. 21 (2015), 297–319; MR 3358544.Google Scholar
Lau, E. and Schleicher, D., Internal addresses in the Mandelbrot set and irreducibility of polynomials, SUNY Stony Brook Preprint 1994/19, arXiv:math/9411238v1.Google Scholar
Looper, N., Dynamical uniform boundedness and the abc-conjecture. Preprint (2019), arXiv:1901.04385v1.CrossRefGoogle Scholar
Merel, L., Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), 437–449 (French); MR 1369424 (96i:11057).CrossRefGoogle Scholar
Morton, P., On certain algebraic curves related to polynomial maps, Compos. Math. 103 (1996), 319–350; MR 1414593.Google Scholar
Morton, P., Galois groups of periodic points, J. Algebra 201 (1998), 401–428; MR 1612390.CrossRefGoogle Scholar
Morton, P., Arithmetic properties of periodic points of quadratic maps. II, Acta Arith. 87 (1998), 89–102; MR 1665198.CrossRefGoogle Scholar
Morton, P. and Silverman, J. H., Rational periodic points of rational functions, Int. Math. Res. Not. IMRN 1994 (1994), 97–110; MR 1264933 (95b:11066).CrossRefGoogle Scholar
Nguyen, K. V. and Saito, M.-H., $d$-gonality of modular curves and bounding torsions, Preprint (1996), arXiv:alg-geom/9603024.Google Scholar
Northcott, D. G., Periodic points on an algebraic variety, Ann. of Math. (2) 51 (1950), 167–177; MR 0034607 (11,615c).CrossRefGoogle Scholar
Ogg, A. P., Hyperelliptic modular curves, Bull. Soc. Math. France 102 (1974), 449–462; MR 0364259 (51 #514).CrossRefGoogle Scholar
Poonen, B., Gonality of modular curves in characteristic p, Math. Res. Lett. 14 (2007), 691–701; MR 2335995.CrossRefGoogle Scholar
Schleicher, D., Internal addresses of the Mandelbrot set and Galois groups of polynomials, Arnold Math. J. 3 (2017), 1–35; MR 3646529.CrossRefGoogle Scholar
Stichtenoth, H., Algebraic function fields and codes, Graduate Texts in Mathematics, vol. 254, second edition (Springer, Berlin, 2009); MR 2464941.Google Scholar
Stoll, M., Rational 6-cycles under iteration of quadratic polynomials, LMS J. Comput. Math. 11 (2008), 367–380; MR 2465796.CrossRefGoogle Scholar