数论讨论班

9月11日 (周五)

15:00–17:00

光华楼东主楼 1601

吉泽坤

(中国科学技术大学)

Title: Periods and de Rham-Betti Groups of Abelian Varieties.

Abstract: The theory of de Rham-Betti (dRB) structures and groups was introduced by André to study polynomial relations among the periods of smooth projective varieties over $\overline{\mathbb{Q}}$, within the framework of Grothendieck's period conjecture. In this talk, I will focus on the case of abelian varieties. Much of our current knowledge about their dRB structures relies on Wüstholz's powerful Analytic Subgroup Theorem. I will explain both the role of this theorem and its limitations in detecting relations among periods. In the second part of the talk, inspired by the work of Kreutz, Shen, and Vial, I will outline a strategy for determining the dRB groups of type IV abelian fourfolds, despite our limited knowledge of the relations among their periods. This talk is based on the preprint https://arxiv.org/abs/2607.23171

9月18日 (周五)

15:00–16:30

光华楼东主楼 2001

袁轶君

(西湖大学)

Title: Formalization of non-Archimedean functional analysis

Abstract: In this talk, I will introduce the formalization of the foundations of non-Archimedean functional analysis in Lean 4. This work includes the basic properties of spherical completeness, examples and non-examples such as the field $\mathbb{C}_p$ of p-adic complex numbers. As applications, we formalize Hahn-Banach extension theorem and the spherical completion for non-Archimedean Banach spaces. If time permits, I will also talk about how formalization strengthen our trust on mathematical proofs in the era of AI.

10月16日 (周五)

16:00–17:00

SCMS 102

林中一攀

(同济大学)

Title: Explicit polynomial presentations of the reduced Emerton-Gee stacks

Abstract: In this talk, I will introduce my program that seeks to (efficiently) compute the reduced Emerton-Gee stacks in non-generic and high dimension situations. I will explain its arithmetic implications such as a speculative Herzig-type Serre weight recipe for non-semisimple, non-generic residual Galois representations and a speculative classification of weight entailment phenomenon. This talk is based on my recent preprint arXiv:2608.01567 (the ``twisted'' Borel case) and the joint work in progress with Min Yu (the reductive case). The resolved twisted Borel cases already imply the existence of crystalline lifts and the qualitative Breuil-Mezard conjecture for most reductive groups (without assumptions on the residual Galois representation).

To orient the audience, I will start with an exposition of the precursor of the Emerton-Gee stacks -- the moduli of potentially Barsotti-Tate rank $2$ Breuil-Kisin modules with tame descent data, and explain how it acquires (twisted) Springer-Schubert geometry. Then I will analyze the non-generic geometry by introducing a mod $p$ ($p=\ell$) version of Weil-Deligne representations.