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

TL;DR

This paper investigates the rational points on a genus 4 curve related to quadratic polynomial dynamics, assuming a conjecture, and combines analytic and algebraic methods to determine the Jacobian's rank and verify the BSD conjecture.

Contribution

It proves the rank of the Jacobian of a specific genus 4 curve is 1 using both conditional and unconditional methods, advancing understanding of rational points in polynomial dynamics.

Findings

The Jacobian rank is 1, confirmed by two independent proofs.

Conditional proof relies on BSD conjecture and L-series conjectures.

Unconditional proof uses class group computations of specific number fields.

Abstract

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, 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 $J$.