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On Poonen’s conjecture concerning rational preperiodic points of quadratic maps. (English) Zbl 1316.37042
This note presents some evidence for B. Poonen’s conjecture [Math. Z. 228, No. 1, 11–29 (1998; Zbl 0902.11025)]: There is no \(c\in \mathbb Q\) such that \(\phi_c(z) = z^2 + c\) has a \(\mathbb Q\)-rational periodic point with exact period greater than \(3\). The authors check this for \(h(c) \leq 8\log 10\) in Proposition 1, where \(h\) is the standard logarithmic height function on \(\mathbb Q\). In Proposition 2, they give an analogous result for a quadratic field \(K\) with discriminant in \([-4000,4000]\): no \(K\)-rational periodic point of exact period greater than \(6\) if \(h(c) \leq 3\log 10\). The method is standard and the same as the computation of torsions on elliptic curves, namely modding out by good-reduction primes and relating the period over finite fields to the period over number fields. The authors reduce the amount of computations by Lemmas 2 and 3, in which they show that a certain ideal has to be a square if there is even a single periodic point in \(K\) for \(\phi_c\). Many reduction-types modulo small primes can also be ruled out, for example if they just have superattracting cycles of small periods. The authors comment that while considering more primes reduces the proportion of reduction-types to check, this increases the amount of computations for each \(c\). They also show in Proposition 4 that the largest exact period of a \(K\)-rational periodic point grows at least linearly in \([K:\mathbb Q]\).

MSC:
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
37P35 Arithmetic properties of periodic points
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