Dynamics of quadratic polynomials and rational points on a curve of genus 4

Abstract

Let f t ( z ) = z 2 + t f_t(z)=z^2+t . For any z ∈ Q z\in \mathbb {Q} , let S z S_z be the collection of t ∈ Q t\in \mathbb {Q} such that z z is preperiodic for f t f_t . In this article, assuming a well-known conjecture of Flynn, Poonen, and Schaefer [Duke Math. J. 90 (1997), pp. 435–463], we prove a uniform result regarding the size of S z S_z over z ∈ Q z\in \mathbb {Q} . In order to prove it, we need to determine the set of rational points on a specific non-hyperelliptic curve C C of genus 4 4 defined over Q \mathbb {Q} . We use Chabauty’s method, which requires us to determine the Mordell-Weil rank of the Jacobian J J of C C . We give two proofs that the rank is 1 1 : an analytic proof, which is conditional on the BSD rank conjecture for J J and some standard conjectures on L-series, and an algebraic proof, which is unconditional, but relies on the computation of the class groups of two number fields of degree 12 12 and degree 24 24 , respectively. We finally combine the information obtained from both proofs to provide a numerical verification of the strong BSD conjecture for J J .