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

Let f_t(z)=z^2+t. For any z\in \mathbb {Q}, let S_z be the collection of t\in \mathbb {Q} such that z is preperiodic for 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 over 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 of genus 4 defined over \mathbb {Q}. We use Chabauty's method, which requires us to determine the Mordell-Weil rank of the Jacobian J of C. We give two proofs that the rank is 1: an analytic proof, which is conditional on the BSD rank conjecture for 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 and degree 24, respectively. We finally combine the information obtained from both proofs to provide a numerical verification of the strong BSD conjecture for J. [ABSTRACT FROM AUTHOR]