Rational points of bounded height on entire curves

TL;DR

This work unifies arithmetic height theory with Nevanlinna theory to bound the number of rational points of bounded height in the image of an entire curve $\varphi: \mathbf{C} \to X(\mathbf{C})$ (with $X$ defined over a number field) inside affine and projective varieties. By relating the Bombieri–Pila counting problem to the Nevanlinna characteristic $T_{\varphi,M}(r)$ and carefully handling base-point dependence and exceptional sets, the authors derive affine and projective Bombieri–Pila-type bounds that are exponential in $H$ and $r$ but sharpen to polynomial growth for many $(r,H)$ via the construction of the sets $L(\varphi,\varepsilon,\gamma)$. The key technical contributions include base-point comparison results, Cartan-type estimates to manage exceptional sets in the projective case, and a Siegel-lemma–based method to produce sections vanishing on prescribed rational points, enabling precise counting. The results have potential Diophantine applications, linking growth properties of entire curves to distribution of rational points on varieties and offering refined tools beyond classical Bombieri–Pila bounds.

Abstract

Let $X$ be an affine or a projective variety defined over a number field $K$ and $\varphi:{\bf C}\to X({\bf C})$ be a holomorphic map with Zariski dense image. We estimate the number of rational points of height bounded by $H$ in the image of a disk of radius $r$ in terms of the the Nevanlinna characteristic function of $\varphi$ and $H$ in a way which generalize the classical Bombieri--Pila estimate to expanding domains. In general this bound is exponential but we show that for many values of $H$ and $r$, the bound is polynomial.

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• Definition 2.2
• Definition 3.1
• Theorem 3.2
• proof