曲线上代数整点的极限分布
Limit distribution of algebraic integral points on curves
- Instituto de Ciencias Matemáticas (ICMAT)（数学科学研究所）
- Beijing International Center for Mathematical Research (BICMR), Peking University（北京大学北京国际数学研究中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究曲线上代数整点伽罗瓦轨道的局部极限分布，推广了Smith和Orloski-Sardari的结果，并证明Szachniewicz定理的整版本及若干应用。
AI中文摘要：
对于一个拟射影算术曲面 $\mathcal U/\mathbb Z$ 以及满足温和正则性假设的紧子集 $E\subset \mathcal U(\mathbb C)$，我们研究 $\mathcal U$ 上伽罗瓦轨道位于 $E$ 中的代数整点。我们刻画了同时作为此类点的伽罗瓦轨道的局部极限分布而出现的局部概率测度族。我们的结果也适用于在部分位点处指定的测度。这推广了 Smith 和 Orloski--Sardari 关于 $\mathcal{U}=\mathbb{A}^1$ 在阿基米德位点处的结果。作为推论，我们证明了 Szachniewicz 定理在曲线上的整版本，该定理指出：每一个整 GVF 泛函都可以在 GVF 拓扑下被一列代数整点逼近。我们还给出了若干应用：我们证明了曲线上高度函数的本质最小值可以由代数整点达到；我们将整数切比雪夫常数与 $\mathbb{A}^1$ 上某些高度函数的本质最小值联系起来，并证明了 Montgomery 的一个猜想；我们还肯定地回答了 Levenberg--Londhe 的一个问题，证明了全正代数整数平均迹的最小极限可以由一列全正代数单位达到。
英文摘要：
For a quasi-projective arithmetic surface $\mathcal U/\mathbb Z$ and a compact subset $E\subset \mathcal U(\mathbb C)$ under mild regularity assumptions, we study algebraic integral points on $\mathcal U$ whose Galois orbits lie in $E$. We characterize the collections of local probability measures that arise simultaneously as the local limit distributions of Galois orbits of such points. Our result also works for measures prescribed at a subset of places. This generalizes the results of Smith and Orloski--Sardari on $\mathcal{U}=\mathbb{A}^1$ concerning the archimedean place. As a consequence, we prove an integral version of Szachniewicz's theorem on curves, which states that every \emph{integral} GVF functional can be approximated by a sequence of algebraic integral points in the GVF topology. We also give several applications: we prove that the essential minimum of height functions on curves can be attained by algebraic integral points; we connect integer Chebyshev constants with the essential minima of certain height functions on $\mathbb{A}^1$ and prove a conjecture of Montgomery; we also answer affirmatively a question of Levenberg--Londhe by showing that the smallest limit of averaged trace of totally positive algebraic integers can be attained by a sequence of totally positive algebraic units.
