发表机构
UCLA; Peking University（加州大学洛杉矶分校; 北京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种源于沃伊塔证明思路的方法，针对整体域上的阿贝尔簇，明确证明了一致有界性猜想与朗-西尔弗曼猜想，还给出特征p>0函数域上椭圆曲线的相关明确界限，并在提出的弱于Hindry版本的高维斯皮罗猜想下对数域证明了这两个猜想。
AI 中文摘要
本文针对定义在整体域K上的阿贝尔簇A，提出一种用于研究一致有界性猜想（Uniform Boundedness Conjecture）与朗-西尔弗曼猜想（Lang-Silverman Conjecture，该猜想给出非挠有理点高度的一致下界）的方法，其思路源于沃伊塔对莫德尔猜想（法尔廷斯定理）的证明。对于特征0的函数域，Looper-Yap的最新突破（arXiv:2603.23396）已证明这两个猜想，但给出的界限不明确；本文给出这两个猜想的新证明，所得界限明确，且除非A/K存在处处好约化的因子，否则界限对域K呈多项式依赖关系。我们还证明了特征p>0的函数域上椭圆曲线的明确界限。对于数域，我们在提出的合适高维斯皮罗猜想（弱于Hindry版本，即该链接的猜想3.4）下证明了这两个猜想。
英文摘要
In this paper, we propose a method to study the {\it Uniform Boundedness Conjecture} and the {\it Lang-Silverman Conjecture} for abelian varieties $A$ defined over a global field $K$; the latter is a uniform lower bound on the heights of non-torsion rational points. Our method is inspired by Vojta's proof of the Mordell Conjecture (Faltings's Theorem). Over function fields of characteristic $0$, a recent breakthrough of Looper-Yap (arXiv:2603.23396) proves both conjectures with inexplicit bounds. In our paper, we give a new proof of both conjectures with explicit bounds, which also depend polynomially on the field $K$ unless $A/K$ admits a factor of good reduction everywhere. We also prove explicit bounds for elliptic curves over function fields of characteristic $p>0$. Over number fields, we prove both conjectures under a suitable high-dimensional Szpiro conjecture (weaker than Hindry's version, Conjecture 3.4 of https://webusers.imj-prg.fr/~marc.hindry/MW-size.pdf) that we propose.
CommentsAdded supercurrents to Section 5.2, and rewrote the proof in Section 5.3 to make it more clear. Corrected other small mistakes and typos. Comments are welcome!
