Arithmetic Algebraic Geometry
in honor of Shouwu Zhang's 60th birthday

 

Event Photos:

 

 

 

 

 

 

Dates: June 5-8, 2023
Speakers:

Miaofen Chen (East China Normal University)

Wee Teck Gan (National University of Singapore)

Xuhua He (Chinese University of Hong Kong)

Yifeng Liu (Zhejiang University)

Ngaiming Mok (University of Hong Kong)

Sian Nie (Academy of Mathematics and Systems Science)

Yannan Qiu (Southern University of Science and Technology)

Ye Tian (Morningside Center of Mathematics)

Yichao Tian (Morningside Center of Mathematics)

Liang Xiao (BICMR, Peking University)

Xinyi Yuan (BICMR, Peking University)

Robin Zhang (Columbia University)

Wei Zhang (Massachusetts Institute of Technology)

Zhiyu Zhang (Massachussetts Institute of Technology)

Organizers: Xuhua He, Michael McBreen, Wei Zhang
Venue:

YIA LT2 (Yasumoto International Academic Park, Room LT2),

Chinese University of Hong Kong.

Information for participants:

Download Campus Map 

IMPORTANT: you must bring photo ID with you to enter the CUHK campus.

There are some discounted rates at the Regal Riverside Hotel for conference participants. Please contact us for more detailed information.

Sponsored by:

CUHK Mathematics Department - https://www.math.cuhk.edu.hk/

Institute of Mathematical Sciences - http://www.ims.cuhk.edu.hk/

New Cornerstone Science Foundation - https://www.newcornerstone.org.cn/#/

 

SCHEDULE

   
Monday (am)
   
9:15-9:30   Opening Speeches
9:30-10:30 Ngaiming Mok Abelian schemes over complex function fields and functional transcendence results
11:00-12:00 Xinyi Yuan Geometric Bombieri-Lang conjecture for finite covers of abelian varieties
Monday (pm)    
2:30-3:30 Wee Teck Gan Relative Langlands duality via Howe duality
4:00-5:00 Robin Zhang Harris-Venkatesh plus Stark
Tuesday (am)
   
9:30-10:30 Ye Tian On Goldfeld conjecture
11:00-12:00 Liang Xiao Slopes of modular forms and ghost conjecture
Tuesday (pm)    
2:30-3:30 Miaofen Chen Harder-Narasimhan stratification in p-adic Hodge theory
4:00-5:00 Yichao Tian An prismatic-etale comparison theorem in the semi-stable case
Wednesday (am)
   
9:30-10:30 Xuhua He Affine Deligne-Lusztig varieties and affine Lusztig varieties  
11:00-12:00 Sian Nie Steinberg's cross-section of Newton strata
Thursday (am)
   
9:30-10:30 Wei Zhang

p-adic Heights of the arithmetic diagonal cycles on unitary Shimura varieties

11:00-12:00 Zhiyu Zhang On Arithmetic Fundamental Lemmas and Arithmetic Transfers
Thursday (pm)    
2:30-3:30 Yannan Qiu About the local functorial lift from a torus
4:00-5:00 Yifeng Liu Anticyclotomic p-adic L-functions for Rankin-Selberg motives

 

ABSTRACTS

Miaofen Chen : Harder-Narasimhan stratification in p-adic Hodge theory

We will talk about the construction of Harder-Narasimhan stratification on the B_{dR}^+-Grassmannian  and study its basic geometric properties, such as non-emptiness, dimension and relation with other stratifications, which  generalizes the work of Dat-Orlik-Rapoport,  Cornut-Peche Irissarry, Nguyen-Viehmann and Shen.  This is a joint work in progress with Jilong Tong.

Wee Teck Gan : Relative Langlands duality via Howe duality

Ben-Zvi, Sakellaridis and Venkatesh recently conjectured that there is a duality phenomenon for the period problems arising in the relative Langlands program, and this should be  the spectral manifestation of a duality of certain Hamiltonian 

varieties with group actions.  In this talk, I will discuss how Howe duality allows one to verify interesting instances of this conjecture.

Xuhua He: Affine Deligne-Lusztig varieties and affine Lusztig varieties   

Roughly speaking, an affine Deligne-Lusztig variety describes the intersection of a given Iwahori double coset with a Frobenius-twisted conjugacy class in the loop group; while an affine Lusztig variety describes the intersection of a given Iwahori double coset with an ordinary conjugacy class in the loop group. The affine Deligne-Lusztig varieties provide a group-theoretic model for the reduction of Shimura varieties and play an important role in the arithmetic geometry and Langlands program. The affine Lusztig varieties encode the information of the orbital integrals of Iwahori-Hecke functions and serve as building blocks for the (conjectural) theory of affine character sheaves. In this talk, I will explain a close relationship between affine Lusztig varieties and affine Deligne-Lusztig varieties, and consequently, proivde an explicit nonemptiness pattern and dimension formula for affine Lusztig varieties in most cases. This talk is based on my preprint arXiv:2302.03203.

Yifeng Liu : Anticyclotomic p-adic L-functions for Rankin-Selberg motives

In this talk, we will construct anticyclotomic p-adic L-functions for motives coming from conjugate-selfdual automorphic Rankin-Selberg products, for both root numbers. We will propose several conjectures concerning such p-adic L-functions and explain certain progress toward one of them, namely, one-side divisibility of a corresponding Iwasawa main conjecture.

Ngaiming Mok : Abelian schemes over complex function fields and functional transcendence results

The speaker has long been interested in applications of complex geometry to number theory, and will trace the trajectory of his involvement revolving around abelian schemes over complex function fields and functional transcendence results on quotients of bounded symmetric domains.

We recall first results of Mok (1991) and Mok-To (1993) concerning the finiteness of Mordell-Weil groups of universal abelian varieties ${\bf A}_\Gamma$ without fixed parts over modular function fields $K = \mathbb C(\overline{X_\Gamma})$.  In these early works in the modular case an invariant K\"ahler form (currently known as the Betti form) was introduced, and, making use of the classifying map, ranks of Mordell-Weil groups were bounded in geometric terms via the volume of the ramification locus.  An important tool was the extendibility of the Betti form as a closed positive current, established in the above works as an  intermediate tool using the methods of P. Lelong and H. Skoda on closed positive currents.

Most recently, Mok-Ng (2022) applied the complex differential geometric approach in the above to prove finiteness results on points of Betti multiplicities $\ge 2$ of a section $\sigma \in {\bf E}(\mathbb C(\overline{X}))$ of infinite order in the case of an elliptic scheme over a quasi-projective curve, a result obtained by Corvaja-Demeio-Masser-Zannier, which was rendered effective by Ulmer-Urz\'ua (2021). Our approach is differential geometric, basing on a fundamental first-order real-linear differential equation satisfied by the verticality $\eta_\sigma$ of  $\sigma \in {\bf E}(\mathbb C(\overline{X}))$, and has the advantage of being applicable in principle to abelian schemes. 

Regarding functional transcendence results we will discuss the Ax-Schanuel theorem of Mok-Pila-Tsimerman (2019) for Shimura varieties and applications of its generalizations to the study of rational points, notably to the proof of the uniform Mordell-Lang theorem of Dimitrov-Gao-Habegger (2021) for number fields, and the characterization of bi-algebraicity due to Chan-Mok (2022) in the case of a projective subvariety $Y \subset X_{\check\Gamma}$, for $X_{\check\Gamma}$ possibly of infinite volume, uniformized by an algebraic subset $Z \subset \Omega$.

Sian Nie : Steinberg's cross-section of Newton strata

Let G be a simply connected semisimple group of rank r over an algebraically closed field. Steinberg has associated to each minimal length Coxeter element an r-dimensional affine space in G, which is a cross-section of all regular conjugacy classes of G. In this talk, we will consider natural analogues of Steinberg’s cross-sections in the context of a loop group equipped with a Frobenius automorphism. We will show how Steinberg’s cross-section intersects Frobenius twisted conjugacy classes (which are parameterized by "Newton polygons"). Some interesting applications will also be discussed.

Yannan Qiu : About the local functorial lift from a torus

Let k be a local field and T a n-dimensional torus in GL_n(k). It is a basic question to explicitly describe or construct the functorial lift from T to GL_n(k). When n=2, such a lift can be constructed via the Weil representation. I will introduce an approach to this question, with emphasis on the case n=3.

Ye Tian : On Goldfeld conjecture

We introduce some recent progress on distribution of 2-Selmer groups and Goldfeld conjecture for quadratic twists of elliptic curves. 

Yichao Tian : An prismatic-etale comparison theorem in the semi-stable case

Various p-adic comparison theorems are important  topics in p-adic Hodge theory. In recent years, the prismatic cohomology theorem introduced by Bhatt and Scholze provide us with a uniform framework to compare various p-adic cohomology theories. In this talk, I will explain a p-adic comparison theorem between the prismatic cohomology for F-crystals and the etale cohomology for local systems on semistable p-adic formal schemes over the ring of integers of a p-adic fields. 

Liang Xiao : Slopes of modular forms and ghost conjecture

The p-adic valuations of the Up eigenvalues of modular forms are called the (p-adic) slopes. The study of this concept was pioneered by the work of Gouvea and Mazur. There has been many interesting conjectures in these subjects, as well as on related topics such as the global geometry of the eigencurve.  In this talk, I will report on a recent joint work with Ruochuan Liu, Nha Truong, and Bin Zhao, in which we proved the so-called ghost conjecture of Bergdall-Pollack, under certain genericity condition. As a corollary, we resolve many of these conjecutres under the same hypothesis.

Xinyi Yuan : Geometric Bombieri-Lang conjecture for finite covers of abelian varieties

The Bombieri-Lang conjecture is a high-dimensional generalization of the Mordell conjecture, and it is a geometric analogue can be formulated over function fields. In a recent joint work, Junyi Xie and I prove the geometric Bombieri-Lang conjecture for finite covers of abelian varieties over function fields of characteristic zero. The goal of this talk is to introduce the result and the idea to prove it. 

Robin Zhang : Harris-Venkatesh plus Stark

The class number formula describes the behavior of the Dedekind zeta function at s=0 and s=1. The Stark and Gross conjectures extend the class number formula, describing the behavior of Artin L-functions and p-adic L-functions at s=0 and s=1 in terms of units and class numbers. The Harris-Venkatesh conjecture describes the residue of Stark units modulo p, giving a modular analogue to the Stark and Gross conjectures while also serving as the first verifiable part of the broader Prasanna-Venkatesh conjectures. In this talk, I will give a picture, formulate a unified conjecture combining Harris-Venkatesh and Stark for weight one modular forms, and describe the proof of this in the imaginary dihedral case.

Wei Zhang : p-adic Heights of the arithmetic diagonal cycles on unitary Shimura varieties.

We formulate a p-adic analogue of the Arithmetic Gan--Gross--Prasad Conjectures for unitary groups, relating the p-adic height pairing of the arithmetic diagonal cycles to the first central derivative (along the cyclotomic direction) of a p-adic Rankin—Selberg L-function associated to cuspidal automorphic representations. In the good ordinary case we are able to prove the conjecture, at least when the ramification are mild at inert primes. We deduce some application to p-adic version of the Bloch-Kato conjecture. This is a joint work with Daniel Disegni. 

Zhiyu Zhang : On Arithmetic Fundamental Lemmas and Arithmetic Transfers

The celebrated Gross-Zagier-Zhang formula relates heights of Heegner points on Shimura curves to central derivatives of L functions of modular forms, which has applications to the BSD conjecture of elliptic curves. As a higher dimensional generalization, the arithmetic Gan-Gross-Prasad (AGGP) conjecture relates heights of diagonal cycles on unitary Shimura varieties to central derivatives of automorphic L functions, which has applications to the Bloch-Kato conjecture of Rankin—Selberg motives. In this talk, I will discuss recent developments on arithmetic fundamental lemmas (AFL) and arithmetic transfers (AT), which in particular lead to the proof of a p-adic AGGP conjecture with mild ramifications by Daniel Disegni and Wei Zhang. And I will discuss my recent work where I formulate and prove the twisted AFL and some ATs, which applies to new type AGGP conjectures and Bloch-Kato conjectures. 

 

 

 

nach oben