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

1. Problem Statement and Context

The paper addresses two interconnected problems in arithmetic geometry and complex analysis:

1. Geometric Problem: Determining the asymptotic growth rate of the minimal Kobayashi length of a non-trivial loop on a bordered hyperbolic Riemann surface when a finite number of points (punctures) are removed.
2. Arithmetic Problem: Improving the quantitative bounds on the number of "generalized integral points" on abelian varieties over complex function fields.

Background:

• Generalized Integral Points: Let $A$ be an abelian variety of dimension $n$ over a function field $K = \mathbb{C}(B)$, where $B$ is a compact Riemann surface. A generalized $(s, B_0)$-integral point is a section $\sigma_P: B \to A$ such that the intersection of its image with a divisor $D$ is contained within a set of at most $s$ disjoint closed disks in $B$.
• Previous Work: Phung [7] established a bound for the number of such points modulo the constant trace. The bound was of the order $O(s^{2nk})$, where $k = \text{rk}(\pi_1(B_0))$. This bound relied on an estimate for the Kobayashi length of loops on the punctured surface $B_0 \setminus S$ (where $|S| \le s$), which Phung proved to be linear, i.e., $O(s)$.
• The Gap: The linear growth of the Kobayashi length led to an exponent of $2nk$ in the counting bound. Phung raised the question of whether the growth of the Kobayashi length is actually slower (specifically, asking if the exponent in the logarithmic growth is $1/2$).

2. Methodology

The paper employs a combination of complex differential geometry (Kobayashi metrics) and arithmetic geometry (Parshin cocycles and lattice counting).

A. Improving the Kobayashi Length Estimate (The Geometric Core)

The author refines the upper bound for the Kobayashi length of a loop $\alpha$ on a surface $B_0$ with $s$ punctures.

• Setup: Let $B_0$ be a bordered hyperbolic surface. For a finite set $S \subset B_0$ with $|S| \le s$, let $\ell_S(\alpha)$ be the minimal Kobayashi length of a loop freely homotopic to $\alpha$ in $B_0 \setminus S$.
• Previous Approach (Phung): Estimated the length of a single specific loop directly, yielding a linear bound $O(s)$.
• New Approach (Dolce):
  1. Fermi Strip Construction: Fix a smooth representative $\gamma$ of $\alpha$ and construct a thin tubular neighborhood (Fermi strip) $T_\delta(\gamma)$ foliated by parallel curves $\gamma_u$.
  2. Integral Averaging: Instead of bounding a single loop, the author bounds the $L^p$-norm of the distortion function $\lambda_S = \kappa_S / \sqrt{\rho}$ (the ratio of the Kobayashi metric to a background Riemannian metric) over the entire strip.
  3. Voronoi Decomposition: The strip is decomposed into Voronoi cells centered at the punctures. Near a puncture, the distortion function behaves like $1/distance$.
  4. Optimization: By choosing an exponent $p$ close to 2 (specifically $p = 2 - 1/\log(s+2)$), the author exploits the fact that $1/r$ is in $L^p$ for $p < 2$. Summing the contributions of $s$ punctures yields a factor of $s/(2-p)$.
  5. Hölder's Inequality: Applying Hölder's inequality converts the $L^p$ bound on the distortion into a bound on the length of a specific parallel curve $\gamma_{u_0}$ that avoids the punctures.
  6. Result: This yields a sublinear bound of $O(\sqrt{s \log s})$.

B. Application to Integral Points (The Arithmetic Core)

The improved geometric bound is substituted into Phung's arithmetic framework.

• Parshin Cocycle: The proof utilizes the Parshin cocycle $c_P: \pi_1(B_0) \to \pi_1(A_b)$, which maps rational points to cohomology classes in $H^1(\pi_1(B_0), \mathbb{Z}^{2n})$.
• Lattice Counting: The number of integral points is bounded by the number of possible cocycles. This reduces to counting lattice points in a ball of radius $R$ in the universal cover of the abelian variety.
• Key Modification:
  • In Phung's proof, the loops used to define the cocycle had to share a common base point, and the length bound was $O(s)$, leading to a displacement bound $R \sim s$. Since the lattice rank is $2n$, the count was $R^{2n} \sim s^{2n}$. With $k$ generators, the total exponent was $2nk$.
  • In Dolce's proof, the new geometric theorem provides loops of length $O(\sqrt{s \log s})$ that do not need to share a common base point.
  • The author shows that even with different base points, the conjugating paths required to compare the loops to a fixed reference have lengths bounded by constants independent of $s$.
  • Consequently, the displacement bound becomes $R \sim \sqrt{s \log s}$.
  • The lattice count becomes $R^{2n} \sim (\sqrt{s})^{2n} = s^n$. With $k$ generators, the total exponent becomes $nk$.

3. Key Contributions and Results

Theorem 1.5 (Geometric Result)

For a non-trivial loop $\alpha$ in a bordered hyperbolic surface $B_0$, the supremum of the minimal Kobayashi length over all sets of $s$ punctures satisfies:
$$L(\alpha, s) \le C \sqrt{(s+1)\log(s+2)}$$
This determines the exact asymptotic growth rate:
$$\lim_{s \to \infty} \frac{\log L(\alpha, s)}{\log s} = \frac{1}{2}$$
This answers Question 1.4 from Phung [7], confirming that the growth is essentially square-root rather than linear.

Theorem 1.4 (Arithmetic Result)

For an abelian variety $A$ of dimension $n$ over $\mathbb{C}(B)$ and a divisor $D$, the number of generalized $(s, B_0)$-integral points modulo the constant trace satisfies:
$$\# \left( I(s, B_0) / \text{Tr}_{K/\mathbb{C}}(A)(\mathbb{C}) \right) \le m (s+1)^{nk + \varepsilon}$$
for every $\varepsilon > 0$.

4. Significance

1. Halving the Exponent: The most significant impact is the reduction of the exponent in the counting bound from $2nk$ to $nk + \varepsilon$. This brings the bound for generalized integral points much closer to the known bounds for standard integral points (where the exponent is typically related to the rank of the Mordell-Weil group, roughly $nk$).
2. Resolution of Open Question: The paper definitively resolves the asymptotic behavior of the Kobayashi length on punctured surfaces, proving the exponent is $1/2$.
3. Methodological Innovation: The introduction of the integral averaging argument over a Fermi strip, combined with Voronoi decomposition and optimized $L^p$ estimates, provides a powerful new tool for estimating geometric quantities in the presence of singularities (punctures).
4. Progress on Lang-Vojta: While the full geometric Lang-Vojta conjecture remains open for general complex abelian varieties, this result provides a significantly sharper quantitative version for the "generalized" setting, narrowing the gap between known results and the conjectured linear bounds.

In summary, Paolo Dolce's work bridges a gap between complex hyperbolic geometry and Diophantine geometry, using a refined geometric estimate to drastically improve the quantitative understanding of integral points on abelian varieties.