Counting algebraic points of bounded degree on curves

TL;DR

The paper investigates algebraic points of bounded degree on a smooth projective curve $X$ over a number field $k$ via a dominant map $f:X\to P^1$ realizing the gonality. By passing to the symmetric power $X^{(\nu)}$ and mapping into the Jacobian $J_X$, the authors relate $f$-rigid points (where $k(x)=k(f(x))$) to $k$-rational points on a subvariety $W_{\nu}\subset J_X$ and connect heights on $X$ to the canonical height on $J_X$. Using Vojta’s finiteness results, Faltings’ theorem, and Néron–Tate height counting on $W_{\nu}$, they show that for odd gonality $\gamma$ and prime $\nu\le\gamma$ there exists $\rho\ge0$ such that the count of $f$-rigid points of degree $\nu$ and height at most $T$ grows like $T^{\rho/2}$. The analysis distinguishes regimes $\nu<\gamma/2$, $\gamma/2<\nu<\gamma$, and $\nu=\gamma$, yielding finite or polynomial-growth behavior and a corollary that infinite sets have at least $T^{1/2}$ growth. The paper also provides explicit examples showing both absence and abundance of $f$-rigid points, illustrating the method’s applicability.

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 $\left\{P\in X\left(\overline{k} \right)\left| [k(x):k]=ν, k(x)=k(f(x)), h(x)\leq T \right.\right\}.$

Theorems & Definitions (14)

• Theorem 1.1
• Definition 1
• Remark 2.1
• Remark 2.2
• Theorem 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• proof