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).
Zhiyou Wu, Daxin Xu, Hongjie Yu
Thursday, 10:30-11:30 am
Venue:
MCM110
Upcoming talks:
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
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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.
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)
Venue: MCM110
Title: A p-adic analogue of a theorem of Narasimhan and Seshadri
Abstract: Given a compact Riemann surface, a classical theorem of Narasimhan and Seshadri characterize vector bundles arising from unitary representations of the fundamental group as the polystable vector bundles of degree 0. Given a projective curve with good reduction over a local field of mixed charateristic 0-p, works of Faltings and Deninger-Werner associate semistable Higgs bundles of degree 0 to representations of the fundamental group. We explain some progress towards the question of characterizing which vector bundles arise in this way.
Date: April 24, 2025
Speaker: Prof. Haoyu Hu (Nanjing University)
Title: Finite push-forward of Abbes-Saito's conductors
Abstract: In this talk, I will introduce a boundedness result of Abbes-Saito's logarithmic and non-logarithmic conductors of an induced representation of an absolute Galois group of a complete discrete valuation field in a geometric situation. I will sketch the proof and an application. This is a joint work with Jean-Baptiste Teyssier.
