发表机构
Morningside Center of Mathematics, Chinese Academy of Sciences; Academy of Mathematics and Systems Science, Chinese Academy of Sciences; University of Chinese Academy of Sciences; Institute for Advanced Study in Mathematics, Harbin Institute of Technology（中国科学院数学与系统科学研究院; 中国科学院数学与系统科学研究院; 中国科学院大学; 哈尔滨工业大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对p进域上的阿贝尔簇，研究刚性解析1-动机理论及阿贝尔簇的共轭单值化，建立了形式与刚性解析1-动机的等价性等结果，还构造了相关p可除群。
AI 中文摘要
设K为p进域，我们研究K上阿贝尔簇的算术理论，研究目标有二。其一，我们研究刚性解析1-动机理论，将其视为描述阿贝尔簇退化的工具，与经典代数情形类似。核心新结果包括：K上形式（或对数形式）1-动机与K上具有好（或半稳定）约化的刚性解析1-动机之间的等价性，以及刚性解析1-动机好（或半稳定）约化的Néron-Ogg-Shafarevich准则；特别地，我们从K上半稳定阿贝尔簇构造出O_K上的对数形式1-动机与对数p可除群。其二，我们研究K上任意阿贝尔簇A的共轭单值化，这是Iovita、Morrow与Zaharescu针对具有好约化的阿贝尔簇所提出的一类p进单值化，本文方法基于Fargues的p可除刚性解析群理论；实际上，我们将共轭单值化理论视为通过Hodge-Tate三元组的分类研究可对偶p可除刚性解析群的有理点，过程中我们从刚性解析1-动机构造出p可除刚性解析群。
英文摘要
Let $K$ be a $p$-adic field. We study the arithmetic theory of abeloid varieties over $K$. Our aims are twofold. First, we study the theory of rigid analytic 1-motives, which will be viewed as a tool to describe degeneration of abeloid varieties, similarly as in the classical algebraic setting. Our key new results are the equivalence between formal (resp. log formal) 1-motives over $\mathcal{O}_K$ and rigid analytic 1-motives with good (resp. semi-stable) reduction over $K$, and the Néron-Ogg-Shafarevich criterion for the good (resp. semi-stable) reduction of rigid analytic 1-motives. In particular, we construct log formal 1-motives and log $p$-divisible groups over $\mathcal{O}_K$ from semi-stable abeloid varieties over $K$. Next, we study the conjugate uniformization of an arbitrary abeloid variety $A$ over $K$. This is a type of $p$-adic uniformization initiated by Iovita--Morrow--Zaharescu in case of abelian varieties with good reduction. Our approach here is based on Fargues' theory of $p$-divisible rigid analytic groups. In fact, we view the theory of conjugate uniformization as a study of rational points of dualizable $p$-divisible rigid analytic groups in terms of their classification Hodge--Tate triples. Along the way, we construct $p$-divisible rigid analytic groups from rigid analytic 1-motives.
Comments104 pages; few corrections and improvements; comments welcome!
