Online Number Theory Seminar

This seminar is held on ZOOM and organized by Morningside Center of Mathematics (MCM) and Yau Mathematical Sciences Center (YMSC).

Organizers

Hansheng Diao (YMSC)

Lei Fu (YMSC)

Yongquan Hu (MCM)

Ye Tian (MCM)

Bin Xu (YMSC)

Weizhe Zheng (MCM)

Conference Room for the current talk

Sechdule

Title: On the locally analytic vectors of the completed cohomology of modular curves

Speaker: Lue Pan (University of Chicago)

Time: 9:30-11:45, September 10, 2020 (Beijing Time)

Abstract: We study locally analytic vectors of the completed cohomology of modular curves and determine eigenvectors of a rational Borel subalgebra of gl_2(Q_p). As applications, we are able to prove a classicality result for overconvergent eigenform of weight one and give a new proof of Fontaine-Mazur conjecture in the irregular case under some mild hypothesis. One technical tool is relative Sen theory which allows us to relate infinitesimal group action with Hodge(-Tate) structure.

Video

Title: Connectedness of Kisin varieties associated to  absolutely irreducible Galois representations

Speaker: Prof. Miaofen Chen (East China Normal University)

Time: 16:00-17:00, August 27, 2020 (Beijing Time)

Abstract: Let K be a p-adic field. Let \rho be a n-dimensional continuous absolutely irreducible mod p representation of the absolute Galois group of K. The Kisin variety is a projective scheme which parametrizes the finite flat group schemes over the ring of integers of K with generic fiber \rho satisfying some determinant condition. The connected components of the Kisin variety is in bijection with the connected components of the generic fiber of the flat deformation ring of \rho with given Hodge-Tate weights.  Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin's conjecture holds if K  is totally ramfied with n=3 or the determinant condition is of a very particular form.  We also give counterexamples to show Kisin's conjecture does not hold in general. This is a joint work with Sian Nie.

Title: Beilinson-Bloch conjecture and arithmetic inner product formula

Speaker: Prof. Yifeng Liu (Yale)

Time: 16:00-17:00, July 23, 2020 (Beijing Time)

Abstract: In this talk, we study the Chow group of the motive associated to a tempered global L-packet \pi of unitary groups of even rank with respect to a CM extension, whose global root number is -1. We show that, under some restrictions on the ramification
of \pi, if the central derivative L'(1/2,\pi) is nonvanishing, then the \pi-nearly isotypic localization of the Chow group of a certain unitary Shimura variety over its reflex field does not vanish. This proves part of the Beilinson--Bloch conjecture for Chow
groups and L-functions. Moreover, assuming the modularity of Kudla's generating functions of special cycles, we explicitly construct elements in a certain \pi-nearly isotypic subspace of the Chow group by arithmetic theta lifting, and compute their heights
in terms of the central derivative L'(1/2,\pi) and local doubling zeta integrals. This is a joint work with Chao Li.

Title: Bound on the number of rational points on curves

Speaker: Prof. Ziyang Gao (CNRS)

Time: 16:00-17:00, July 9, 2020 (Beijing Time)

Abstract: Mazur conjectured, after Faltings's proof of the Mordell conjecture, that the number of rational points on a curve depends only on the genus, the degree of the number field and the Mordell-Weil rank. This conjecture was established in a few cases. In this talk I will explain how to prove this conjecture and some of its generalization. I will focus on how functional transcendence and unlikely intersections on mixed Shimura varieties are applied. This is joint work with Vesselin Dimitrov and Philipp Habegger.

Title: Towards three problems of Katz on Kloosterman sums

Speaker: Prof. Ping Xi (Xi'an Jiaotong University )

Time: 16:00-17:00, June 24, 2020 (Beijing Time)

Abstract: Motivated by deep observations on elliptic curves, Nicholas Katz proposed three problems on sign changes, equidistributions and modular structures of Kloosterman sums in 1980. In this talk, we will discuss some recent progresses towards these three problems made by analytic number theory combining certain tools from $\ell$-adic cohomology.

Title: Algebraic cycles on Shimura varieties and L-functions

Speaker: Prof. Wei Zhang (MIT)

Time: 9:30-11:00, June 11, 2020 (Beijing Time)

Abstract: This will be an introductory talk to special algebraic cycles on Shimura varieties and their relation to L-functions. No prior knowledge Shimura varieties will be assumed.

PPT & Video

Title: The arithmetic fundamental lemma for p-adic fields

Speaker: Prof. Wei Zhang (MIT)

Time: 9:00-10:30, June 4, 2020 (Beijing Time)

Abstract: The arithmetic fundamental lemma (AFL) is a conjectural identity relating the arithmetic intersection numbers on a Rapoport-Zink space for unitary groups to the first derivative of relative orbital integral on the general linear groups over a p-adic field F. The AFL was proved in the case F=Q_p about one year ago. In this talk I will report a work in progress joint with A. Mihatsch to prove the AFL for a general p-adic field.

PPT & Video