Rational 6-Cycles Under Iteration of Quadratic Polynomials

Abstract

We present a proof, which is conditional on the Birch and Swinnerton-Dyer Conjecture for a specific abelian variety, that there do not exist rational numbers x and c such that x has exact period N = 6 under the iteration x ↦ x2 + c. This extends earlier results by Morton for N = 4 and by Flynn, Poonen and Schaefer for N = 5.