Density of algebraic points on products of curves

TL;DR

This paper studies the distribution of algebraic points on surfaces, especially products of curves, revealing different behaviors depending on degree and providing explicit examples with applications to algebraic geometry.

Contribution

It introduces a systematic analysis of algebraic point density on surfaces, highlighting the influence of degree and arithmetic properties, with new explicit examples and applications.

Findings

Density behavior varies with degree and arithmetic properties.

Explicit examples of products of genus 2 curves show different density patterns.

Applications include insights into rank growth on abelian and bielliptic surfaces.

Abstract

In this paper, we initiate the systematic study of density of algebraic points on surfaces. We give an effective asymptotic range in which the density degree set has regular behavior dictated by the index. By contrast, in small degree, the question of density is subtle and depends on the arithmetic of the curves. We give several explicit examples displaying these different behaviors, including products of genus $2$ curves with and without dense quadratic points. These results for products of curves have applications to questions about algebraic points on closely related surfaces, such as rank growth on abelian surfaces and bielliptic surfaces.