Kobayashi length bounds on bordered surfaces and generalized integral points on abelian varieties

TL;DR

This paper refines bounds on Kobayashi lengths on bordered surfaces and improves the counting estimates for generalized integral points on abelian varieties over complex function fields.

Contribution

It sharpens the upper bound of Kobayashi length growth to $O( oot{s ext{log}s})$ and halves the exponent in counting generalized integral points.

Findings

Established the limit $rac{ ext{log} L( ext{alpha},s)}{ ext{log} s} = rac{1}{2}$.

Improved the counting bound for generalized integral points to $s^{nk+ ext{epsilon}}$.

Sharpened previous bounds on Kobayashi length growth and integral point counting.

Abstract

Let $B$ be a compact Riemann surface and $B_0\subset B$ a bordered hyperbolic subsurface obtained by removing finitely many disjoint closed disks. Fix a nontrivial loop $\alpha$ in $B_0$. For $s\ge 0$, let $L(\alpha,s)$ denote the supremum, over all finite subsets $S\subset B_0$ with $\#S\le s$, of the minimal Kobayashi length of a loop in $B_0\smallsetminus S$ that is freely homotopic to $\alpha$ in $B_0$. Phung in [7] proved that $L(\alpha,s)$ grows at most linearly and at least as $\sqrt{s}/\log s$. We sharpen the upper bound to $O\left(\sqrt{s\log s}\right)$, which determines $\lim_{s\to\infty}\frac{\log L(\alpha,s)}{\log s}=\frac{1}{2}$, answering a question raised in [7, Question 1.4]. As an application, we improve the counting bound for generalized integral points on abelian varieties over complex function fields: for an abelian variety of dimension $n$ over $\mathbb C(B)$, Phung proved that the number of $(s, B_0)$-generalized integral points modulo the constant trace grows at most as $s^{2nk}$, where $k=\operatorname{rk}(\pi_1(B_0))$. We sharpen this to $s^{nk+\varepsilon}$ for every $\varepsilon>0$, halving the exponent.