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Cycles of quadratic polynomials and rational points on a genus-2 curve. (English) Zbl 0958.11024

Let \(f\) be a quadratic polynomial with rational coefficients. It was proved by P. Morton [Acta Arith. 87, 89-102 (1998)] that \(f\) cannot have rational periodic points of period 4 and the authors show that the period \(5\) also cannot occur. Since, as noted by Morton, the pairs \((f,P)\) with \(P\) being a rational periodic point of period \(N\) are classified by an algebraic curve \(C_1(N)\), the problem is reduced to the study of rational points on \(C_1(5)\), which is of degree 30, has genus 14 and is not modular (Theorem 7).
The authors consider a quotient curve \(C_0(5)\) of \(C_1(5)\), which is of genus 2, transform it in a hyperelliptic form and compute (via \(2\)-descent) the rank of its Jacobian, which happens to be 1. This allows them to bound the number of rational points on \(C_0(5)\), using a refinement of the method of C. Chabauty [C. R. Acad. Sci., Paris 212, 882-885 (1941; Zbl 0025.24902)] and R. F. Coleman [Duke Math. J. 52, 765-770 (1985; Zbl 0588.14015)]. It results that \(C_0(5)\) has exactly six rational points and this suffices to prove the assertion about rational periodic points of period 5.
The authors conjecture that for \(N\geq 4\) there is no rational quadratic polynomial having a rational periodic point of period \(N\) and give a numerical argument to support their conjecture in the case \(N=6\).

MSC:

11C08 Polynomials in number theory
14E05 Rational and birational maps
37B99 Topological dynamics
14G05 Rational points
11D41 Higher degree equations; Fermat’s equation
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[1] T. Bousch, Sur quelques problèmes de dynamique holomorphe , thèse, Universitè de Paris-Sud, Centre d’Orsay, 1992.
[2] G. Call and J. Silverman, Canonical heights on varieties with morphisms , Compositio Math. 89 (1993), no. 2, 163-205. · Zbl 0826.14015
[3] J. W. S. Cassels, Local fields , London Mathematical Society Student Texts, vol. 3, Cambridge University Press, Cambridge, 1986. · Zbl 0595.12006
[4] J. W. S. Cassels, The Mordell-Weil group of curves of genus \(2\) , Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Mass., 1983, pp. 27-60. · Zbl 0529.14015
[5] C. Chabauty, Sur les points rationnels des courbes algébriques de genre supérieur à l’unité , C. R. Acad. Sci. Paris 212 (1941), 882-885. · Zbl 0025.24902
[6] R. F. Coleman, Effective Chabauty , Duke Math. J. 52 (1985), no. 3, 765-770. · Zbl 0588.14015 · doi:10.1215/S0012-7094-85-05240-8
[7] E. V. Flynn, The Jacobian and formal group of a curve of genus \(2\) over an arbitrary ground field , Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 3, 425-441. · Zbl 0723.14023 · doi:10.1017/S0305004100068729
[8] E. V. Flynn, The group law on the Jacobian of a curve of genus \(2\) , J. Reine Angew. Math. 439 (1993), 45-69. · Zbl 0765.14014 · doi:10.1515/crll.1993.439.45
[9] E. V. Flynn, Descent via isogeny in dimension \(2\) , Acta Arith. 66 (1994), no. 1, 23-43. · Zbl 0835.14009
[10] E. V. Flynn, An explicit theory of heights , Trans. Amer. Math. Soc. 347 (1995), no. 8, 3003-3015. JSTOR: · Zbl 0864.11033 · doi:10.2307/2154766
[11] E. V. Flynn, A flexible method for applying Chabauty’s theorem , Compositio Math. 105 (1997), no. 1, 79-94. · Zbl 0882.14009 · doi:10.1023/A:1000111601294
[12] D. Gordon and D. Grant, Computing the Mordell-Weil rank of Jacobians of curves of genus two , Trans. Amer. Math. Soc. 337 (1993), no. 2, 807-824. JSTOR: · Zbl 0790.14028 · doi:10.2307/2154244
[13] D. Grant, Formal groups in genus two , J. Reine Angew. Math. 411 (1990), 96-121. · Zbl 0702.14025 · doi:10.1515/crll.1990.411.96
[14] N. Katz, Galois properties of torsion points on abelian varieties , Invent. Math. 62 (1981), no. 3, 481-502. · Zbl 0471.14023 · doi:10.1007/BF01394256
[15] M. Kenku, On the number of \({\mathbf Q}\)-isomorphism classes of elliptic curves in each \({\mathbf Q}\)-isogeny class , J. Number Theory 15 (1982), no. 2, 199-202. · Zbl 0493.14017 · doi:10.1016/0022-314X(82)90025-7
[16] A. Knapp, Elliptic curves , Mathematical Notes, vol. 40, Princeton University Press, Princeton, NJ, 1992. · Zbl 0804.14013
[17] S. Lang, Abelian varieties , Interscience Tracts in Pure and Applied Mathematics. No. 7, Interscience Publishers, Inc., New York, 1959. · Zbl 0098.13201
[18] H. Lange, Kurven mit rationaler Abbildung , J. Reine Angew. Math. 295 (1977), 80-115. · Zbl 0384.14007 · doi:10.1515/crll.1977.295.80
[19] B. Levi, Saggio per una teoria aritmetica della forme cubiche ternarie , Atti Accad. Reale Sci. Torino 43 (1908), 99-120.
[20] D. Lewis, Invariant sets of morphisms on projective and affine number spaces , J. Algebra 20 (1972), 419-434. · Zbl 0245.12003 · doi:10.1016/0021-8693(72)90067-1
[21] W. G. McCallum, On the Shafarevich-Tate group of the Jacobian of a quotient of the Fermat curve , Invent. Math. 93 (1988), no. 3, 637-666. · Zbl 0661.14033 · doi:10.1007/BF01410203
[22] W. G. McCallum, The arithmetic of Fermat curves , Math. Ann. 294 (1992), no. 3, 503-511. · Zbl 0766.14013 · doi:10.1007/BF01934338
[23] L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres , Invent. Math. 124 (1996), no. 1-3, 437-449. · Zbl 0936.11037 · doi:10.1007/s002220050059
[24] J. Merriman and N. Smart, Curves of genus \(2\) with good reduction away from \(2\) with a rational Weierstrass point , Math. Proc. Cambridge Philos. Soc. 114 (1993), no. 2, 203-214. · Zbl 0805.14018 · doi:10.1017/S030500410007153X
[25] J. S. Milne, Jacobian varieties , Arithmetic geometry (Storrs, Conn., 1984) eds. G. Cornell and J. H. Silverman, Springer, New York, 1986, pp. 167-212. · Zbl 0604.14018
[26] P. Morton, Arithmetic properties of periodic points of quadratic maps, II , preprint, 1995. · Zbl 1029.12002
[27] P. Morton, On certain algebraic curves related to polynomial maps , Compositio Math. 103 (1996), no. 3, 319-350. · Zbl 0860.11065
[28] P. Morton and J. Silverman, Rational periodic points of rational functions , Internat. Math. Res. Notices (1994), no. 2, 97-110. · Zbl 0819.11045 · doi:10.1155/S1073792894000127
[29] P. Morton and J. Silverman, Periodic points, multiplicities, and dynamical units , J. Reine Angew. Math. 461 (1995), 81-122. · Zbl 0813.11059 · doi:10.1515/crll.1995.461.81
[30] W. Narkiewicz, On polynomial transformations in several variables , Acta Arith. 11 (1965), 163-168. · Zbl 0148.41801
[31] D. Northcott, Periodic points on an algebraic variety , Ann. of Math. (2) 51 (1950), 167-177. JSTOR: · Zbl 0036.30102 · doi:10.2307/1969504
[32] B. Poonen, Torsion in rank-\(1\) Drinfel’d modules and the uniform boundedness conjecture , to appear in Math. Ann. · Zbl 0891.11034 · doi:10.1007/s002080050090
[33] E. Pyle, Abelian varieties over \(\mathbb{Q}\) with large endomorphism algebras and their simple components over \(\bar{\mathbb{Q}}\) , Ph.D. thesis, Univ. of Calif., Berkeley, 1995.
[34] K. Ribet, Endomorphism algebras of abelian varieties attached to newforms of weight \(2\) , Seminar on Number Theory, Paris 1979-80, Progr. Math., vol. 12, Birkhäuser Boston, Mass., 1981, pp. 263-276. · Zbl 0467.14006
[35] K. Ribet, Abelian varieties over \({\mathbf Q}\) and modular forms , Algebra and topology 1992 (Taejŏn), Korea Adv. Inst. Sci. Tech., Taejŏn, 1992, pp. 53-79. · Zbl 1092.11029
[36] E. F. Schaefer, \(2\)-descent on the Jacobians of hyperelliptic curves , J. Number Theory 51 (1995), no. 2, 219-232. · Zbl 0832.14016 · doi:10.1006/jnth.1995.1044
[37] R. Walde and P. Russo, Rational periodic points of the quadratic function \(Q_ c(x)=x^ 2+c\) , Amer. Math. Monthly 101 (1994), no. 4, 318-331. JSTOR: · Zbl 0804.58036 · doi:10.2307/2975624
[38] W. C. Waterhouse and J. S. Milne, Abelian varieties over finite fields , 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math. Soc., Providence, R.I., 1971, pp. 53-64. · Zbl 0216.33102
[39] T. Youssefi, Inégalité relative des genres , Manuscripta Math. 78 (1993), no. 2, 111-128. · Zbl 0812.14016 · doi:10.1007/BF02599304
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