志村簇的几何和算术应用

We plan to hold a specialized research-level workshop on geometry of Shimura varieties and arithmetic applications, and apply to Tianyuan Foundation for financial support. The study of geometry of Shimura varieties is a key link in the Langlands program, and it has many arithmetic applications. Many Chinese scholars, domestic or not, have done leading works in this area worldwide. For example, the three applicants and Xinwen Zhu gave a global characterization of the basic locus of the special fiber of Shimura varieties, and Yifeng Liu, Yichao Tian, Liang Xiao, Wei Zhang, and Xinwen Zhu used the geometry of Shimura varieties to give new cases for Beilinson-Bloch-Kato conjecture. We hope to gather experts from China and worldwide to discuss key questions in the area of geometry of Shimura varieties, and to discuss future directions of development. On the one hand, our event targets global experts and postdocs, in hope to form a coherent and robust-discussing research group; on the other hand, we will invite good domestic PhD students, in hope to inspire their interest in the study of geometry of Shimura varieties, and possibly introduce them to relevant topics.

我们计划举办关于志村簇几何和算术应用的高级研讨班，并申请天元基金对这个项目进行资助。对志村簇几何的研究是朗兰兹纲领中重要一环，它有着很多算术应用。很多国内学者和华裔学者在这方面的工作在国际上都处于领先位置，例如三位申请人和朱歆文在对志村簇特殊纤维基本区域的整体刻画，以及刘一峰、田一超、肖梁、张伟和朱歆文利用志村簇几何对Beilinson-Bloch-Kato猜想给出新例证等等。我们希望借此研讨班聚集国内和国际专家共同探讨关键问题的新进展和未来发展方向。我们的活动一方面以国内外专家和博士后为主，希望建立一个相互了解、积极讨论的学术圈，另一方面我们也将邀请国内优秀博士生共同参加，希望可以激起他们对研究志村簇几何的兴趣，并投入到相关课题的研究中。

朗兰兹纲领是数学中最核心和最受关注的课题之一。它将不同的数学分支相连，包括数论、代数几何、表示论和调和分析。志村簇的几何可以被用来研究算术不变量和解析不变量直接深刻的关系。最近在这一方向上有一些不错的进展，包括证明Beilinson-Bloch-Kato猜想的一些特例，从而将BSD猜想推广到高维情形。..我们组织会议和高级研讨班邀请相关方向的专家参加并寻求新的研究方向。同时，我们还特别注重打造一支由年轻人组成的有凝聚力的有强大科研能力的队伍。