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

Flynn, E. V.; Poonen, Bjorn; Schaefer, Edward F.

Duke Math. J. 90, No. 3, 435-463 (1997).

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 \(N=6\).

Reviewer: Władisław Narkiewicz (Wrocław)

MSC:

11C08	Polynomials in number theory
14E05	Rational and birational maps
37B99	Topological dynamics
14G05	Rational points
11D41	Higher degree equations; Fermat’s equation

Keywords:

periodic points; quadratic polynomials; hyperelliptic curves

Citations:

Zbl 0025.24902; Zbl 0588.14015

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