Counting Rational Points on Cubic Curves

TL;DR

This paper establishes uniform upper bounds for the number of rational points on non-singular cubic curves over the rationals, using a novel combination of the determinant method and m-descent, with bounds depending on the Jacobian's rank.

Contribution

It introduces a new approach combining the determinant method with m-descent to obtain uniform bounds depending on the Jacobian's rank.

Findings

Established uniform upper bounds for rational points

Bound depends on the Jacobian's rank

Method combines determinant method with m-descent

Abstract

We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals. The bounds are uniform in the curve and involve the rank of the corresponding Jacobian. The method used in the proof is a combination of the "determinant method" with an m-descent on the curve.