Bounded rational points on curves

TL;DR

This paper provides a precise estimate for the number of rational points of bounded height on irreducible projective curves, extending to hypersurfaces with coefficient-sensitive results.

Contribution

It establishes the sharp estimate <<_d N^{2/d} for rational points on curves and general hypersurfaces, improving understanding of point distribution based on geometric and algebraic properties.

Findings

Sharp estimate for rational points on curves

Coefficient-sensitive results for hypersurfaces

Extension of point counting techniques

Abstract

We establish the sharp estimate <<_d N^{2/d} for the number of rational points of height at most N on an irreducible projective curve of degree d. We deduce this from a result for general hypersurfaces that is sensitive to the coefficients of the corresponding form.