Uniform Bounds for the Number of Rational Points on Symmetric Squares of Curves with Low Mordell-Weil Rank

TL;DR

This paper establishes conditional uniform bounds on the number of rational points on symmetric squares of curves with low Mordell-Weil rank, extending previous bounds for curves to their symmetric squares.

Contribution

It provides the first uniform bounds for rational points on symmetric squares of curves with low Mordell-Weil rank, improving upon prior results for curves.

Findings

Conditional bounds depend on the rank of the Jacobian

Bounds are uniform outside the algebraic special set

Rank-favorable bounds are obtained for hyperelliptic cases

Abstract

A central problem in Diophantine geometry is to uniformly bound the number of $K$-rational points on a smooth curve $X/K$ in terms of $K$ and its genus $g$. A recent paper by Stoll proved uniform bounds for the number of $K$-rational points on a hyperelliptic curve $X$ provided that the rank of the Jacobian of $X$ is at most $g - 3$. Katz, Rabinoff and Zureick-Brown generalized his result to arbitrary curves satisfying the same rank condition. In this paper, we prove conditional uniform bounds on the number of rational points on the symmetric square of $X$ outside its algebraic special set, provided that the rank of the Jacobian is at most $g-4$. We also find rank-favorable uniform bounds (that is, bounds depending on the rank of the Jacobian) in the hyperelliptic case.