On the global determinant method

TL;DR

This paper explicitly develops Salberger's global determinant method using Arakelov geometry and applies it to analyze how the number of rational points of bounded height on plane curves depends on their degree.

Contribution

It provides an explicit construction of Salberger's global determinant method via Arakelov geometry and explores the impact of the Generalized Riemann Hypothesis on constants involved.

Findings

Explicit bounds on rational points depending on degree

Analysis of constants under the Generalized Riemann Hypothesis

Enhanced understanding of the distribution of rational points

Abstract

In this paper, we build the global determinant method of Salberger by Arakelov geometry explicitly. As an application, we study the dependence on the degree of the number of rational points of bounded height in plane curves. We will also explain why some constants will be more explicit if we admit the Generalized Riemann Hypothesis.