AAAAA Seminar

AAAAA Seminar
2025-04-03 10:30-11:30
MCM110

Introduction:

This is a series seminar organized by MCM on Algebraic / Analytic number theory, Arithmetic geometry, Algebraic groups and Automorphic forms at MCM110 on each Thursday. If you have any questions, please feel free to contact us (mcmoffice@math.ac.cn). 


Organizers:

Zhiyou Wu, Daxin Xu, Hongjie Yu

 

Time:

Thursday, 10:30-11:30 am

 

Venue:

MCM110

 

Upcoming talks:

Date: April 3, 2025

Speaker: Prof. Xu Shen (MCM, CAS)

Venue: MCM110

Title: Bruhat-Tits buildings and p-adic period domains

Abstract: Bruhat-Tits buildings and p-adic period domains are both basic objects associated to p-adic reductive groups. In this talk, we will discuss some basic relations between them. This is joint work with Ruishen Zhao.


Date: April 10, 2025

Speaker: Yuchan Lee (POSTECH)

Venue: MCM410

Title: An asymptotic formula for the number of integral matrices with a fixed characteristic polynomial via orbital integrals

Abstract: For an arbitrarily given irreducible polynomial χ(x) in Z[x] of degree n, let N (X, T ) be the number of n × n matrices over Z whose characteristic polynomial is χ(x), bounded by a positive number T with respect to a certain norm.
We provide an asymptotic formula for N (X, T ) as T → ∞ in terms of the orbital integrals of gln. This generalizes the work of A. Eskin, S. Mozes, and N. Shah (1996) which assumed that Z[x]/(χ(x)) is the ring of integers.

In addition, we will provide an asymptotic formula for N (X, T ), using the orbital integrals of gln, when Q is generalized to a totally real number field k and when n is a prime number. Here we need a mild restriction on splitness of χ(x) over kv at p-adic places v of k for p ≤ n when k[x]/(χ(x)) is unramified Galois over k. This is a joint work with Seongsu Jeon.


Date: April 17, 2025

Speaker: Prof. Fabrizio Andreatta (Università Statale di Milano)

Title: TBA

Abstract: TBA


Date: April 24, 2025

Speaker: Prof. Haoyu Hu (Nanjing University)

Title: TBA

Abstract: TBA


Date: May 8, 2025

Speaker: Dr. Andreas Mihatsch (Universitat Bonn)

Title: TBA

Abstract: TBA


Date: May 15, 2025

Speaker: Prof. Nadir Matringe (NYU Shanghai)

Title: TBA

Abstract: TBA

 

Date: May 22, 2025

Speaker: Prof. Qirui Li (POSTECH)

Title: TBA

Abstract: TBA


Date: May 29, 2025

Speaker: Prof. Jiangtao Li (Central South University)

Title: TBA

Abstract: TBA


Date: June 5, 2025

Speaker: Prof. Yang Cao (Shandong University)

Title: TBA

Abstract: TBA

 

Date: June 12, 2025

Speaker: Dr. Fei Chen (YMSC)

Title: TBA

Abstract: TBA

 

Date: June 19, 2025

Speaker: TBA

Title: TBA

Abstract: TBA 

--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

 

Date: February 27, 2025

Speaker: Ruishen Zhao (MCM, CAS)

Title: p-adic Level Rasing on the Eigenvariety of Definite U(3)

Abstract: Classical level raising theory (developed by Ribet, Taylor etc) is mainly about modulo p congruence between cuspidal eigenforms. Later James Newton considered p-adic analogue for overconvergent automorphic forms on definite quaternion algebras. In this talk I will generalize his results to definite unitary groups, which can also be seen as a p-adic analogue of classical level raising for definite U(3) by Bellaiche and Graftieaux. As a motivation, I will first illustrate a local analogue for GL(n), which relates degenerate principal series to intersection points on the moduli space of tame L-parameters. Then we turn to global p-adic setting. After introducing basic notations, I prove a kind of abelian Ihara lemma for any definite unitary groups U(n). After that we specify to n = 3 case. Use a natural pairing and some duality arguments, we deduce some level raising results about support of some Hecke modules coming from p-adic automorphic forms. This result will imply some intersection points on the eigenvariety.

 

Date: March 6, 2025

Speaker: Prof. Zhizhong Huang (AMSS, CAS)

Title: Arithmetic purity of strong approximation and sieve methods

Abstract: We say that a nice variety over a number field satisfies strong approximation if the set of rational points is dense in the adelic space. A problem first proposed by O. Wittenberg asks whether this property holds true for any given Zariski open subset whose complement has codimension at least two. We shall present some quanlitative and quantitative positive answers in this direction. Our method combines effective counting results from homogeneous dynamics and various sieve methods, e.g. the affine linear sieve (developed by P. Sarnak et al.) and the Ekedahl geometric sieve. This is based on joint work in progress with Y. Cao (Jinan) and R. Zhang (Chongqing).

 

Date: March 13, 2025

Speaker: Dr. Qijun Yan (BIMSA)

Title: On the reductions of integral Frobenius period maps for Shimura varieties

Abstract: Thanks to recent work by Imai, Kato, and Youcis on the prismatic realization of Shimura varieties, one can construct a Breuil-Kisin prismatic period map for the p-integral model of a Shimura variety with good reduction at p. With that construction, we study the reductions of this integral map and derive a mod p Frobenius period map for the corresponding mod p Shimura variety (denoted by S). This new map can be regarded as an enhancement of the zip period map for S.

 

Date: March 20, 2025

Speaker: Dr. Arnaud Vanhaecke (MCM) 

Title: An Alternative Proof of Drinfeld's Representability Theorem

Abstract: The p-adic symmetric space and its formal model have been of fundamental importance in many arithmetic geometry topics. One of the main reasons is its realization as a moduli space of deformations of certain p-divisible groups, known as special formal O_D-modules. This fact, called Drinfeld's theorem, was sketched by Drinfeld in the 70s in an 8-page paper and worked out  in the rank 1 case by Boutot and Carayol over more than 60 pages. Using modern tools in p-adic geometry, we propose an alternative proof of Drinfeld's theorem. This is joint work with Sebastian Bartling.

 

Date: March 27, 2025

Speaker: Prof. Heer Zhao (HIT)

Title: Tame covers and Kummer log flat torsors (joint with J. Gillibert)

Abstract: Let X be a regular scheme, D a normal crossing divisor on X, and U the complement of D. We endow X with the canonical log structure associated to D. Let G be a finite flat group scheme over X. We discuss the relations among tame G-covers of X relative to D, fppf G-torsors over U, and Kummer log flat G-torsors over X.