Counting rational points on smooth cubic curves

TL;DR

This paper employs a global $p$-adic determinant method to establish uniform upper bounds on the number of rational points of bounded height on smooth cubic curves over $Q$, depending only on the Jacobian's rank.

Contribution

It introduces a uniform upper bound for rational points on smooth cubic curves using a global $p$-adic determinant method, extending previous local approaches.

Findings

Upper bounds depend only on the Jacobian's rank

Bounds are uniform across all non-singular cubic curves over $Q$

Method leverages Salberger's adaptation of Heath-Brown's $p$-adic determinant approach

Abstract

We use a global version of Heath-Brown's $p-$adic determinant method developed by Salberger to give upper bounds for the number of rational points of height at most $B$ on non-singular cubic curves defined over $\mathbb{Q}$. The bounds are uniform in the sense that they only depend on the rank of the corresponding Jacobian.