Counting algebraic points of bounded degree on curves

TL;DR

This paper investigates how the number of algebraic points of bounded degree and height on a curve grows, especially those points that are images of points with a fixed degree under a gonality-realizing morphism.

Contribution

It provides new bounds and insights into the distribution and growth rate of algebraic points of bounded degree on algebraic curves, linking gonality and point counting.

Findings

Establishes growth rate estimates for algebraic points of bounded degree.

Connects gonality of the curve with the distribution of algebraic points.

Provides bounds on the number of points with fixed degree and height constraints.

Abstract

Let $X$ be a smooth projective curve over a number field $k$. Let $f\colon X \to \mathbb{P}^1$ be a non-constant morphism that realizes the gonality of $X$. In this article we study the growth rate of $\left\{P\in X\left(\overline{k} \right)\left| [k(x):k]=\nu , k(x)=k(f(x)), h(x)\leq T \right.\right\}.$