2016 Workshop on Number Theroy

Aug. 2 – 4, 2016, Morningside Center of Mathematics

Organizer: Prof. Ye Tian (MCM, CAS)

Place: Room 817 of South Building of AMSS

Speakers:

Ø Hang Xue (Max Planck Institute for Mathematics)

Schedule:

Title & Abstract:

     Title:  An RTF identity for the global arithmetic Gan-Gross-Prasad conjecture

Speaker:  Wei Zhang (Columbia University)

Abstract: I will state a conjectural RTF identity (here RTF stands for "relative trace formula" in the sense of Jacquet) arising from the global arithmetic Gan-Gross-Prasad conjecture for unitary Shimura varieties. This is a joint work with B. Smithling and M. Rapoport.

Title:  Compactifications of splitting models of PEL-type Shimura varieties

Speaker:  Kai-Wen Lan (University of Minnesota)

Abstract: I will report on some constructions of toroidal and minimal compactifications, with expected properties concerning stratifications and formal local structures, for integral models of PEL-type Shimura varieties defined by taking normalizations over the splitting models considered by Pappas and Rapoport.

Title: Algebraic and tropical de Rham theories for non-Archimedean analytic spaces

Speaker: Yifeng Liu (Northwestern University)

Abstract: We show that algebraic de Rham cohomology sheaves of smooth non-Archimedean analytic spaces have canonical decomposition via local weights. This in particular confirms a question raised by Berkovich about 1-forms. On the other hand, we study tropical Dolbeault cohomology in the sense of Chambert-Loir and Ducros, and will show a tight connection between these two theories. Some applications and conjectures will also be discussed.

      Title: Rapoport-Zink spaces of abelian type

Speaker: Xu Shen (AMSS, CAS)

Abstract: We will report some work in progress on the generalized Rapoport-Zink spaces, which appear in the local aspect of our work on the geometry of perfectoid Shimura varieties of abelian type. We will also discuss some applications to K3 surfaces and their moduli.

      Title:  On ghost conjecture of Bergdall and Pollack

Speaker: Liang Xiao (University of Connecticut, Storrs)

Abstract: In this talk, I will report on a recent emerging area studying the slopes of modular forms and slopes of exponential sums. These two topics share a very interesting and surprising feature, namely, there is a clean but non-trivial description of the slopes. I will summarize the main results and explain some potential future directions.  This involves joint works with Chris Davis, Ruochuan Liu, Rufei Ren, Daqing Wan, Myungjun Yu, and Jun Zhang.

       Title:  Applications of Bertolini-Darmon-Prasanna Formula

 Speaker:  Xin Wan (AMSS, CAS)

 Abstract:  I'll talk about two applications of the BDP formula -- the converse of Gross-Zagier-Kolyvagin theorem and the BSD formula in the rank one case.

      Title: Polarized isogeny orbits in families of abelian varieties

 Speaker: Ziyang Gao (CNRS and Princeton University)

 Abstract: Let $\mathcal{A}$ be an abelian scheme over $B$. Fix a point $s$ in $\mathcal{A}$ and let $\Sigma$ be the polarized isogeny orbit generated by $s$. We characterize curves in $\mathcal{A}$ whose intersections with $\Sigma$ is Zariski dense and give the conjecture for higher dimensional subvarieties.

      Title: On the representation theory of Brylinski-Deligne extensions

Speaker: Wen-Wei Li (AMSS, CAS)

Abstract: Weissman proposed a program to study the covers of reductive groups arising from the K_2 extensions constructed by Brylinski-Deligne, and conjectured a local Langlands correspondence for genuine representations of these covers. For reductive groups, it is well-known that L-packets are intimately related to stable conjugacy, but little is known about covering groups. I will try to explore these issues and propose a notion of stable conjugacy for even coverings of Sp(2n), which seems to be new.

      Title: Refined global Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on symplectic groups

Speaker: Hang Xue (Max Planck Institute for Mathematics)

Abstract: We propose a conjectural identity between the Fourier--Jacobi periods on symplectic groups and the central value of certain Rankin--Selberg $L$-functions. We explain how this identity can be deduced from the Ichino--Ikeda's conjecture in some cases.

      Title: Rigidity of p-adic local systems and application to Shimura varieties

Speaker: Ruochuan Liu (Peking University)

Abstract: I will report some recent progress on de Rham rigidity of p-adic local systems as well as its application to Shimura varieties.

     Title: proetale crystalline site and comparison theorems

Speaker: Yichao Tian (AMSS, CAS)

Abstract:  Let X be a proper smooth scheme over the ring of integers of  a finite extension of Q_p. Classical crystalline comparison theorem relates the crystalline cohomology of the special fiber of X to the p-adic etale cohomology of its generic fiber. All the previous results seem to either require some extra conditions on the ramification of the base ring, or work only for constant coefficients. In this work, we propose an approach to crystalline comparison theorem that should work for any proper smooth formal schemes defined over O_{C_p} and allow non-constant coefficients.  The key ingredient is a new site that interpolates both the crystalline site and Scholze’s proetale site. This is a joint work in progress with Jilong Tong.

      Title: Künneth formula for nearby cycles (after Illusie)

Speaker: Weizhe Zheng (AMSS, CAS)

Abstract: We present Illusie's work on the local convolution product of vanishing cycles, analogue of the classical Thom-Sebastiani theorem. One main ingredient is a Künneth formula for nearby cycles over general bases.

      Title: Integral crystalline cohomology and Kisin modules

Speaker: Tong Liu (Purdue University)

Abstract: In this talk, I will discuss the how to construct integral comparison theorem (to compare integral crystalline cohomology and integral etale cohomlogy), when K is ramified and Hodge-Tate weight is small. In particular, I will discuss how integral crystalline cohomology relates to various modules studied by Breuil and Kisin to classify lattices in crystalline representation. This is collaborated work with Bryden Cais.

      Title:  Algebraic dynamics of the lifts of Frobenius

Speaker:  Junyi Xie (Institut de Mathématiques de Toulouse)

Abstract:  In this talk we study some questions in the algebraic dynamics for endomorphisms of projective spaces with coefficients in a p-adic field whose reduction in positive characteritic is the Frobenius.  Our method is based on the theory of perfectoid spaces introduced by P. Scholze.

      Title:  automorphic periods, motivic periods and special values of L-functions. 

Speaker: Jie Lin (IHES)

Abstract: The Deligne conjecture relates special values of a motivic L-function and the Deligne period. When the motive is associated to the tensor product of two automorphic representations, we may reformulate this conjecture in terms of automorphic periods. In this talk, we will present this reformulation and introduce some known results and ongoing work for the reformed conjecture.