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

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 $α$ in $B_0$. For $s\ge 0$, let $L(α,s)$ 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 $α$ in $B_0$. Phung in [7] proved that $L(α,s)$ 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 $\lim_{s\to\infty}\frac{\log L(α,s)}{\log s}=\frac{1}{2}$, 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 $k=\operatorname{rk}(π_1(B_0))$. We sharpen this to $s^{nk+\varepsilon}$ for every $\varepsilon>0$, halving the exponent.

Figures (1)

• Figure 1: The Fermi strip $T_\delta(\gamma)$ around $\gamma$, with punctures $p_1,\ldots,p_5\in S$ and the parallel curve $\gamma_u$.

Theorems & Definitions (18)

• Conjecture 1.1: Geometric Lang--Vojta
• Definition 1.2
• Theorem 1.3: Phung Phung
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Corollary 2.4
• proof