召集人：李时璋（中国科学院数学与系统科学研究院晨兴数学中心，副研究员）、申旭，中国科学院数学与系统科学研究院晨兴数学中心，研究员）、Koji Shimizu，（清华大学丘成桐数学科学中心，助理教授）

时间：2025.03.09—2025.03.15

TIANYUAN WORKSHOP: RECENT ADVANCES IN P-ADIC GEOMETRY 

Date: March 10th – 14th, 2025

Location: Tianyuan Mathematics Research Center, Yunnan Province, China 

Organizers: 

Shizhang Li (Morningside Center of Mathematics, Chinese Academy of Sciences) 

Xu Shen (Morningside Center of Mathematics, Chinese Academy of Sciences) 

Koji Shimizu (Yau Mathematical Sciences Center, Tsinghua University) 

Purpose & Goal: In the past decade, p-adic geometry has been developing rapidly and become an important and active field in arithmetic geometry and number theory. This workshop aims to invite experts, both in China and overseas, discuss some of the latest developments in p-adic geometry, and promote future research and collaborations. We adopt an Arbeitsgemeinschaft (study group) approach and study three preprints [Sch24, DRC24, RC24] in the workshop. There are also short talks by junior participants to facilitate communication and discussions. 

Schedule: Evenings and Wednesday afternoon are left for discussions. 

Topics: 

Berkovich Motives [Sch24, Sch25]: This lecture series discusses Scholze’s theory on Berkovich motives and its application to the geometrization of the local Langlands correspondence. We plan to follow [Sch24] in detail since it is relatively self-contained. The last two talks briefly discuss [Sch25]. 

Talk1 Koji Shimizu (Tsinghua University) Give an overview and review prior works. [Sch24, §1] 

Talk2 Yigeng Zhao (Westlake University) Briefly review Banach rings and Berkovich spectrum. Then discuss the arc topology on Banach rings. [Sch24, §2, 3] 

Talk3 Yihang Zhu (Tsinghua University) Discuss the stable ∞-category of finitary arc-sheaves. Briefly recall the relevant definitions about ∞-categories if necessary. [Sch24, §4] 

Talk4 Heer Zhao (Harbin Institute of Technology) Discuss the stable ∞-category of effective motivic sheaves. [Sch24, §5] 

Talk5 Naoki Imai (The University of Tokyo) Discuss free motivic sheaves and the cancellation theorem. Organize your talk so that the key ideas are clearly presented. [Sch24, §6, 7] 

Talk6 Enlin Yang (Peking University) Compare K-theory and motivic cohomology rationally. Keep in mind that many participants are not experts in K-theory and review relevant backgrounds if necessary. [Sch24, §8] 

Talk7 Weizhe Zheng (Chinese Academy of Sciences) Discuss the symmetric monoidal presentable stable ∞-category of motivic sheaves. [Sch24, §9, 10] 

Talk8 Koji Shimizu (Tsinghua University) Compare Scholze’s theory with Voevodsky’s. If time permits, review some backgrounds for [Sch25]. [Sch24, §11, 12] 1

Talk9 Heng Du (Tsinghua University) Discuss the application to the geometrization of the local Langlands correspondence. [Sch25] 

Jackquet–Langlands [DRC24]: This lecture series discusses the locally analytic theory of infinite level local Shimura varieties and its application to p-adic Jacquet–Langlands functor. We assume that the participants are familiar with the Fargues–Fontaine curve and p-adic period sheaves. 

Talk1 Xu Shen (Chinese Academy of Sciences) Give an overview. [DRC24, §1] 

Talk2 Benchao Su (Peking University) Discuss solid locally analytic representations. Then briefly recall basics on solid almost quasi-coherent sheaves. [DRC24, §2.4, 2.2] 

Talk3 Arnaud Vanhaecke (Chinese Academy of Sciences) Review the basic theory on local Shimura varieties. [DRC24, §3, 2.5] 

Talk4 Wansu Kim (Korea Advanced Institute Science and Technology) Discuss the theory of geometric Sen operators. [RC22a, Theorems 2.4.3, 3.3.4] 

Talk5 Xu Shen (Chinese Academy of Sciences) Review the computation of geometric Sen operators in the global setting of Shimura varieties to provide necessary motivations for the local setting. Discuss geometric Sen operators on local Shimura varieties. [RC22b, Theorems 5.1.4, 5.2.5, 6.3.5] and [DRC24, §4] 

Talk6 Yong Suk Moon (Beijing Institute of Mathematical Sciences and Applications) Discuss locally analytic vectors in the cohomology of towers of rigid spaces. [DRC24, §5.1] 

Talk7 Hui Gao (Southern University of Science and Technology) Compare compactly supported de Rham cohomologies of two towers in a duality of local Shimura varieties. Discuss the locally analytic Jacquet-Langlands functor in the Lubin-Tate case. [DRC24, §5.2, 5.3] 

De Rham Stacks [RC24]: The goal of this lecture series is to give a detailed overview of Rodr´ıguez Camargo’s theory on analytic de Rham stacks. The first talk will purposely omit or distort technical details, and the participants who are interested in this topic are encouraged to read [RC24] directly. 

Talk1 Shizhang Li (Chinese Academy of Sciences) We shall motivate with a “classical” interpretation of D-modules, then move to introduce necessary background in analytic geometry a la Clausen–Scholze, lastly we end up with introducing analytic (de Rham) stacks. [RC24, §4,5] 

Talk2 Shanwen Wang (Renmin University of China) Discuss the application to analytic D-modules. [RC24, §6] 

Short Talks Short talks are 20 minutes long and scheduled after the regular talks. The goal is to let junior participants introduce their research interests and directions to others. 

Talk1 Khai-Hoan Nguyen-Dang (Padova University) 

Talk2 Shengkai Mao (Chinese Academy of Sciences) 

Talk3 Xiangqian Yang (Peking University) 

References

[DRC24] Gabriel Dospinescu and Juan Esteban Rodr´ıguez Camargo, A Jacquet-Langlands functor for p-adic locally analytic representations, arXiv:2411.17082. 

[RC22a] Juan Esteban Rodr´ıguez Camargo, Geometric Sen theory over rigid analytic spaces, arXiv:2205.02016v3. 

[RC22b] , Locally analytic completed cohomology, arXiv:2209.01057v2. 

[RC24] , The analytic de Rham stack in rigid geometry, arXiv:2401.07738. 

[Sch24] Peter Scholze, Berkovich Motives, arXiv:2412.03382. 

[Sch25] , Geometrization of the local Langlands correspondence, motivically, arXiv:2501.07944. 2