Counting Rational Points on Cubic Curves

2009/09/23 by D. R. Heath‐Brown, Heath-Brown, D. R., Damiano Testa +1 · 1 citation
Computer Science · Mathematics · #11D25 #11D45 #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.0909.4246

openalex publication_date 2009/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• Counting rational points on smooth quartic and quintic surfaces

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