### Online Number Theory Seminar

##### 2022-04-07 16:00-17:00

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)

Sechdule

TitleHodge-Tate prismatic crystals and Sen theory

SpeakerProf. Hui Gao (SUSTech)

Time16:00-17:00, April 7, 2022 (Beijing time)

PlaceZOOM (Zoom ID: 466 356 2952  Passcode: mcm1234

AbstractLet K be a p-adic field with perfect residue field. We show the category of rational Hodge-Tate prismatic crystals on O_K is equivalent to the category of nearly Hodge-Tate representations. This proves a conjecture of Yu Min and Yupeng Wang. As a key ingredient of the proof, we reconstruct Sen theory using a Kummer tower (instead of the classical cyclotomic tower).

Title: Growth of Bianchi modular forms

SpeakerDr. Weibo Fu (Princeton University)

Time: 9:00-10:00, March 3, 2022 (Beijing time)

Place: ZOOM (Zoom ID: 466 356 2952  Passcode: mcm1234

Abstract: In this talk, I will establish a sharp bound on the growth of cuspidal Bianchi modular forms. By the Eichler-Shimura isomorphism, we actually give a sharp bound of the second cohomology of a hyperbolic three manifold (Bianchi manifold) with local system arising from the representation Sym^k \otimes \overline{Sym^k} of SL_2(C). I will explain how a p-adic algebraic method is used for deriving our result.

Title: Linear Arithmetic Fundamental Lemma and Intersection numbers for CM cycles on Lubin—Tate spaces

SpeakerDr. Qirui Li (University of Toronto)

Time: 9:00-10:30, June 3, 2021 (Beijing time)

Place: MCM110 & ZOOM  (Zoom ID: 466 356 2952  Passcode: mcm1234

AbstractThe Guo-Jacquet Fundamental Lemma is a higher dimensional generalization of the local field analogue of the Waldspurger formula. It has an arithmetic generalization called the Linear Arithmetic Fundamental Lemma. It is conjectured by Wei Zhang interpreting the derivative of certain orbital integral into certain intersection number of Lubin-Tate spaces, which is a local analogue of the Gross-Zagier formula. We will introduce the known results for the linear Arithmetic Fundamental Lemma, and the intersection number formula for Lubin-Tate spaces. After a joint work with Ben Howard, we also discovered a bi-quadratic generalization of the conjecture.

Title Adelic line bundles over quasi-projective varieties and the uniform Mordell-Lang problem

SpeakerProf. Xinyi Yuan (Peking University)

Time: 15:30-17:30, May 20, 2021  (Beijing Time)

Place: MCM110 & ZOOM  (Zoom ID: 466 356 2952  Passcode: mcm1234

AbstractThis talk consists of two parts. The first part sketches a theory of adelic line bundles and equidistribution over quasi-projective varieties, introduced in my recent joint work with Shou-Wu Zhang. The second part talks about my recent treatment of the uniform Mordell-Lang problem using adelic line bundles, which extends the breakthrough of Dimitrov-Gao-Habegger and Kuhne on this problem to base fields of all characteristics by a different method.

Title Cohomological Descent for Faltings' p-adic Hodge Theory and Applications

SpeakerTongmu He (Paris-Saclay University & IHES)

Time: 16:00-17:00, May 13, 2021  (Beijing Time)

PlaceZOOM  (Zoom ID: 6920686706295  Passcode: 763270

AbstractFaltings' approach in p-adic Hodge theory can be schematically divided into two main steps: firstly, a local reduction of the computation of the p-adic étale cohomology of a smooth variety over a p-adic local field to a Galois cohomology computation and then, the establishment of a link between the latter and differential forms. These relations are organized through Faltings ringed topos whose definition relies on the choice of an integral model of the variety, and whose good properties depend on the (logarithmic) smoothness of this model. Scholze's generalization for rigid analytic varieties has the advantage of depending only on the variety (i.e. the generic fibre). Inspired by Deligne's approach to classical Hodge theory for singular varieties, we establish a cohomological descent result for the structural sheaf of Faltings topos, which makes it possible to extend Faltings' approach to any integral model, i.e. without any smoothness assumption. An essential ingredient of our proof is a descent result of perfectoid algebras in the arc-topology due to Bhatt and Scholze. As an application of our cohomological descent, using a variant of de Jong's alteration theorem for morphisms of schemes, we generalize Faltings' main p-adic comparison theorem to any proper and finitely presented morphism of coherent schemes over an absolute integral closure of Z_p (without any assumption of smoothness) for torsion étale sheaves (not necessarily finite locally constant).

Title On the EKOR stratifications for reductions of Shimura varieties with parahoric level

SpeakerProf. Chao Zhang (Shing-Tung Yau Center of Southeast University)

Time: 16:30-17:30, Jan. 14, 2021  (Beijing Time)

Abstract:  I will present basic results as well as basic ideas for EKOR strata. We will mainly talk about Hodge type cases where the group is tamely ramified at p together with a certain restriction on its fundamental group. We will also indicate how to generalize the results to Shimura varieties of abelian type, if time permits. This talk is based on a joint work with Xu Shen and Chia-Fu Yu.

Title: Big images of Galois representations associated to Hida families

SpeakerDr. Huan Chen(Huazhong University Of Science And Technology)

Time: 9:00-10:00, Dec 03, 2020 (Beijing Time)

Abstract: We study the images of Galois representations associated to Hida families. In 2015, Hida and Jacyln Lang have proved that the image of Ga-lois representation associated to a non-CM family of ordinary classical modular forms contains a congruence subgroup of Iwasawa algebra of weights and also of a nite extension of this algebra, which is called the self-twist ring. Hida and Jacques Tilouine have generalized the results about the existence of nontrivial congruence subgroups contained in the image of the Galois representation to the case of a Hida family of "general" Siegel modular forms of genus 2. Under some technical hypothesis, we have proved the same result for general reductive groups, when the associated Galois representation exists. In this talk, I will explain the results and show the main steps of the proof. This is a joint work with Jacques Tilouine.

Title: Generalized special cycles and theta series

SpeakerDr. Yousheng Shi (University of Wisconsin-Madison)

Time: 9:00-10:00, October 22, 2020 (Beijing Time)

Abstract: We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using differential forms with values in the Weil representation, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.

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

Title: On certain special values of $L$-functions associated to elliptic curves and real quadratic fields

Speaker: Prof. Chung Pang Mok (Soochow University)

Time: 16:30-17:30September 24, 2020 (Beijing Time)

Abstract: We study a class of normalized special values of $L$-functions associated to elliptic curves (defined over $\mathbf{Q}$) and real quadratic fields. Under certain hypotheses, we are able to show that these are squares of rational numbers. This result is used to improve upon a theorem of Bertolini-Darmon about a $p$-adic Gross-Zagier type formula.