A local–global principle in the dynamics of quadratic polynomials

Abstract

Let [Formula: see text] be a number field, [Formula: see text] a quadratic polynomial, and [Formula: see text]. We show that if [Formula: see text] has a point of period [Formula: see text] in every non-archimedean completion of [Formula: see text], then [Formula: see text] has a point of period [Formula: see text] in [Formula: see text]. For [Formula: see text] we show that there exist at most finitely many linear conjugacy classes of quadratic polynomials over [Formula: see text] for which this local–global principle fails. By considering a stronger form of this principle, we strengthen global results obtained by Morton and Flynn–Poonen–Schaefer in the case [Formula: see text]. More precisely, we show that for every quadratic polynomial [Formula: see text] there exist infinitely many primes [Formula: see text] such that [Formula: see text] does not have a point of period four in the [Formula: see text]-adic field [Formula: see text]. Conditional on knowing all rational points on a particular curve of genus 11, the same result is proved for points of period five.