Dynamics of quadratic polynomials and rational points on a curve of genus $4$
HTML articles powered by AMS MathViewer
- by Hang Fu and Michael Stoll;
- Math. Comp. 93 (2024), 397-410
- DOI: https://doi.org/10.1090/mcom/3883
- Published electronically: July 20, 2023
- HTML | PDF | Request permission
Abstract:
Let $f_t(z)=z^2+t$. For any $z\in \mathbb {Q}$, let $S_z$ be the collection of $t\in \mathbb {Q}$ such that $z$ is preperiodic for $f_t$. In this article, assuming a well-known conjecture of Flynn, Poonen, and Schaefer [Duke Math. J. 90 (1997), pp. 435–463], we prove a uniform result regarding the size of $S_z$ over $z\in \mathbb {Q}$. In order to prove it, we need to determine the set of rational points on a specific non-hyperelliptic curve $C$ of genus $4$ defined over $\mathbb {Q}$. We use Chabauty’s method, which requires us to determine the Mordell-Weil rank of the Jacobian $J$ of $C$. We give two proofs that the rank is $1$: an analytic proof, which is conditional on the BSD rank conjecture for $J$ and some standard conjectures on L-series, and an algebraic proof, which is unconditional, but relies on the computation of the class groups of two number fields of degree $12$ and degree $24$, respectively. We finally combine the information obtained from both proofs to provide a numerical verification of the strong BSD conjecture for $J$.References
- E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932, DOI 10.1007/978-1-4757-5323-3
- Raymond van Bommel, David Holmes, and J. Steffen Müller, Explicit arithmetic intersection theory and computation of Néron-Tate heights, Math. Comp. 89 (2020), no. 321, 395–410. MR 4011549, DOI 10.1090/mcom/3441
- Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
- Nils Bruin, Bjorn Poonen, and Michael Stoll, Generalized explicit descent and its application to curves of genus 3, Forum Math. Sigma 4 (2016), Paper No. e6, 80. MR 3482281, DOI 10.1017/fms.2016.1
- Xander Faber, Benjamin Hutz, and Michael Stoll, On the number of rational iterated preimages of the origin under quadratic dynamical systems, Int. J. Number Theory 7 (2011), no. 7, 1781–1806. MR 2854215, DOI 10.1142/S1793042111004162
- G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366 (German). MR 718935, DOI 10.1007/BF01388432
- E. V. Flynn, Bjorn Poonen, and Edward F. Schaefer, Cycles of quadratic polynomials and rational points on a genus-$2$ curve, Duke Math. J. 90 (1997), no. 3, 435–463. MR 1480542, DOI 10.1215/S0012-7094-97-09011-6
- Wilfred W. J. Hulsbergen, Conjectures in arithmetic algebraic geometry, Aspects of Mathematics, E18, Friedr. Vieweg & Sohn, Braunschweig, 1992. A survey. MR 1150049, DOI 10.1007/978-3-322-85466-7
- V. A. Kolyvagin, Finiteness of $E(\textbf {Q})$ and SH$(E,\textbf {Q})$ for a subclass of Weil curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 3, 522–540, 670–671 (Russian); English transl., Math. USSR-Izv. 32 (1989), no. 3, 523–541. MR 954295, DOI 10.1070/IM1989v032n03ABEH000779
- The LMFDB Collaboration, The L-functions and Modular Forms Database, 2022. http://www.lmfdb.org.
- Patrick Morton, Arithmetic properties of periodic points of quadratic maps. II, Acta Arith. 87 (1998), no. 2, 89–102. MR 1665198, DOI 10.4064/aa-87-2-89-102
- Patrick Morton and Joseph H. Silverman, Rational periodic points of rational functions, Internat. Math. Res. Notices 2 (1994), 97–110. MR 1264933, DOI 10.1155/S1073792894000127
- Bjorn Poonen, The classification of rational preperiodic points of quadratic polynomials over $\textbf {Q}$: a refined conjecture, Math. Z. 228 (1998), no. 1, 11–29. MR 1617987, DOI 10.1007/PL00004405
- Bjorn Poonen and Michael Stoll, The Cassels-Tate pairing on polarized abelian varieties, Ann. of Math. (2) 150 (1999), no. 3, 1109–1149. MR 1740984, DOI 10.2307/121064
- Edward F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve, Math. Ann. 310 (1998), no. 3, 447–471. MR 1612262, DOI 10.1007/s002080050156
- Michael Stoll, Independence of rational points on twists of a given curve, Compos. Math. 142 (2006), no. 5, 1201–1214. MR 2264661, DOI 10.1112/S0010437X06002168
- Michael Stoll, Rational 6-cycles under iteration of quadratic polynomials, LMS J. Comput. Math. 11 (2008), 367–380. MR 2465796, DOI 10.1112/S1461157000000644
- Michael Stoll, Magma code verifying the computational claims in this paper, 2022-03-24. available at https://www.mathe2.uni-bayreuth.de/stoll/magma/genus4curve2022.magma.
- John Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1995, pp. Exp. No. 306, 415–440. MR 1610977
Bibliographic Information
- Hang Fu
- Affiliation: Department of Mathematics, National Taiwan University, Taipei, Taiwan
- MR Author ID: 1172768
- ORCID: 0000-0002-1214-4664
- Email: drfuhang@gmail.com
- Michael Stoll
- Affiliation: Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany
- MR Author ID: 325630
- ORCID: 0000-0001-5868-2962
- Email: Michael.Stoll@uni-bayreuth.de
- Received by editor(s): July 17, 2022
- Received by editor(s) in revised form: March 20, 2023
- Published electronically: July 20, 2023
- © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 93 (2024), 397-410
- MSC (2020): Primary 11G30, 11G40, 14G05, 14G10, 14H45, 37P05
- DOI: https://doi.org/10.1090/mcom/3883
- MathSciNet review: 4654627