数学研究所
数学科学全国重点实验室
华罗庚青年数学论坛报告
Speaker: Dr. Tongmu He (Princeton University)
Inviter: 许大昕 副研究员
Title: Introduction to Sen theory I: what are Sen operators?
Time&Venue: 2025年5月21日（星期三） 9:30-10:30 & 晨兴110
Abstract:Sen theory is a foundational approach to nonabelian p-adic Hodge theory. Introduced by Shankar Sen in 1980, it provides a systematic framework to study continuous representations of the absolute Galois group of the p-adic field Q_p. In this talk, I will introduce the core ideas of Sen theory, and explain the construction and significance of the Sen operator. These ideas remain central and continue to thrive in today’s vibrant field of p-adic geometry. Undergraduates and graduate students familiar with basic algebraic number theory are very welcome!
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Title: Introduction to Sen theory II: relation to p-adic Simpson correspondence.
Time&Venue: 2025年5月21日（星期三） 11:00-12:00 & 晨兴110
Abstract:Sen theory, originally developed for studying Galois representations over p-adic local fields, has recently been extended to the representations of the étale fundamental groups of smooth varieties. A parallel and deeply related approach is the p-adic Simpson correspondence, which associates Higgs bundles to such representations. In this talk, I will introduce and compare these two foundational frameworks in nonabelian p-adic Hodge theory.
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Title: Introduction to Sen theory III: perfectoidness criteria.
Time&Venue: 2025年5月22日（星期四） 14:00-15:30 & 晨兴110
Abstract: A longstanding question in the theory of Shimura varieties concerns their perfectoidness at infinite level—a property that would reveal deep connections between étale and coherent cohomology. In this talk, we establish a criterion for perfectoidness via Sen theory, building on a new development of p-adic Hodge theory for general valuation fields that extends Tate’s foundational work on local fields. We further provide a conceptual explanation, based on the p-adic Simpson correspondence after Abbes-Gros, Liu-Zhu and Tsuji, for why Shimura varieties satisfy this criterion, at least in the case of modular curves. For general Shimura varieties, it follows through additional technical arguments due to Pan and Rodríguez Camargo. This yields the “pointwise perfectoidness” of Shimura varieties at infinite level, which suffices to establish the desired connection between different cohomologies. As an application, we show that integral completed cohomology groups vanish in higher degrees, thereby confirming a conjecture of Calegari and Emerton for arbitrary Shimura varieties.
