Abstract:

Many important arithmetic geometric/Diophantine problems are about abelian varieties and/or use abelian varieties as important tools. The recent development of p-adic Hodge theory sometimes allows one to go beyond abelian varieties and/or bypass using abelian varieties. Some examples include Scholze’s construction of moduli local Shtukas, Liu-Zhu, and Diao-Lan-Liu-Zhu’s work on general Shimura varieties, Lawrence-Venketesh’s new proof of the Mordell conjecture, and the most proof of the full Andre-Oort conjecture by Pila-Shankar-Tsimerman. We propose to have a seminar to study an important ingredient in this last work, namely the construction of canonical heights on general Shimura varieties. In the future, we can also move to other topics, if participants have interests and energy.

Time: Every Friday 10-11:30 am starting on Oct. 22nd, 2021

10/22

Xinwen Zhu (California Institute of Technology) : Overview

11/05

Hui Gao (Sustech)

Title: Heights of Motives

Abstract: I will mostly follow Koshikawa's paper "On height of motives with semistable reduction" (since PST paper only needs crystalline motives). I will also briefly mention Kato's paper "Height functions for motives".

11/12

Xu Daxin (MCM)

Title: Relative Fontaine-Laffaille functor

Abstract: I will talk about Faltings' relative Fontaine-Laffaille functor from the category of filtered modules to the category of Z_p local systems and an enhancement due to Tsuji.

11/19

Xu Daxin (MCM)

Title: The p-adic Riemann-Hilbert correspondence due to Liu-Zhu

Abstract: I will review the p-adic Riemann-Hilbert correspondence developed by Liu-Zhu and compares it with relative Fontaine-Laffaille functor. If time permits, we will also review the work of Diao-Lan-Liu-Zhu in the logarithmic case.

11/26

no seminar

12/3

Ruochuan Liu (PKU)

Title: Frobenius structures on rigid connections

Abstract: I will explain the appendix of [PST]. The main result is that under certain condtions, rigid connections can be equipped with a Frobenius structure, thus becoming a relative Fontaine-Laffaille module.

12/10, 12/17, 12/31

Liang Xiao (PKU)

Title：Shimura varieties and special points

Abstract: In this talk, I will give a quick run down of the basics of Shimura varieties and assoicated local systems, and vector bundles. Also, we discuss the special points and partial CM types following Pila-Shanka-Tsimerman Sections 3 and 4.

12/31

Yihang Zhu (Maryland)

Title: Comparison of two constructions of automorphic vector bundles

Abstract: The p-adic Riemann-Hilbert correspondence in [DLLZ] provides a way to construct automorphic vector bundles on Shimura varieties by applying the correspondence to p-adic automorphic local systems. We discuss the theorem in loc. cit. that this construction agrees with the classical complex analytic construction. Along the way we introduce an important ingredient:  Margulis’ super-rigidity.

01/07（start at 11am）, 01/11, 01/14 

Yihang Zhu (Maryland)

01/18, 01/22

Kai-Wen Lan (UMN)

Title: Norms on automorphic vector bundles

Abstract: We explain the construction in [PST] of admissible collections of norms on the Hodge graded pieces of automorphic vector bundles, and their application to the definition of heights.

01/25, 01/28

Xinyi Yuan (PKU)

Title: Recent developments on the Andre-Oort conjecture

Abstract: The goal of these talks are to introduce the proof of the Andre--Oort conjecture for the Siegel case by Tsimerman et al and for the full case by Pila-Shankar-Tsimerman.

Zoom ID: 812 8543 7322

Password: 202473