Arithmetic Algebraic Geometry
in honor of Shouwu Zhang's 60th birthday
Dates:  June 58, 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: 
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. 
SCHEDULE 

Monday (am) 

9:159:30  Opening Speeches  
9:3010:30  Ngaiming Mok  Abelian schemes over complex function fields and functional transcendence results 
11:0012:00  Xinyi Yuan  Geometric BombieriLang conjecture for finite covers of abelian varieties 
Monday (pm)  
2:303:30  Wee Teck Gan  Relative Langlands duality via Howe duality 
4:005:00  Robin Zhang  HarrisVenkatesh plus Stark 
Tuesday (am) 

9:3010:30  Ye Tian  On Goldfeld conjecture 
11:0012:00  Liang Xiao  Slopes of modular forms and ghost conjecture 
Tuesday (pm)  
2:303:30  Miaofen Chen  HarderNarasimhan stratification in padic Hodge theory 
4:005:00  Yichao Tian  An prismaticetale comparison theorem in the semistable case 
Wednesday (am) 

9:3010:30  Xuhua He  Affine DeligneLusztig varieties and affine Lusztig varieties 
11:0012:00  Sian Nie  Steinberg's crosssection of Newton strata 
Thursday (am) 

9:3010:30  Wei Zhang 
padic Heights of the arithmetic diagonal cycles on unitary Shimura varieties 
11:0012:00  Zhiyu Zhang  On Arithmetic Fundamental Lemmas and Arithmetic Transfers 
Thursday (pm)  
2:303:30  Yannan Qiu  
4:005:00  Yifeng Liu  Anticyclotomic padic Lfunctions for RankinSelberg motives 
ABSTRACTS 
Miaofen Chen : HarderNarasimhan stratification in padic Hodge theory We will talk about the construction of HarderNarasimhan stratification on the B_{dR}^+Grassmannian and study its basic geometric properties, such as nonemptiness, dimension and relation with other stratifications, which generalizes the work of DatOrlikRapoport, CornutPeche Irissarry, NguyenViehmann and Shen. This is a joint work in progress with Jilong Tong. 
Wee Teck Gan : Relative Langlands duality via Howe duality BenZvi, 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 DeligneLusztig varieties and affine Lusztig varieties Roughly speaking, an affine DeligneLusztig variety describes the intersection of a given Iwahori double coset with a Frobeniustwisted 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 DeligneLusztig varieties provide a grouptheoretic 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 IwahoriHecke 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 DeligneLusztig 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 padic Lfunctions for RankinSelberg motives In this talk, we will construct anticyclotomic padic Lfunctions for motives coming from conjugateselfdual automorphic RankinSelberg products, for both root numbers. We will propose several conjectures concerning such padic Lfunctions and explain certain progress toward one of them, namely, oneside 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 MokTo (1993) concerning the finiteness of MordellWeil 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 MordellWeil 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, MokNg (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 quasiprojective curve, a result obtained by CorvajaDemeioMasserZannier, which was rendered effective by UlmerUrz\'ua (2021). Our approach is differential geometric, basing on a fundamental firstorder reallinear 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 AxSchanuel theorem of MokPilaTsimerman (2019) for Shimura varieties and applications of its generalizations to the study of rational points, notably to the proof of the uniform MordellLang theorem of DimitrovGaoHabegger (2021) for number fields, and the characterization of bialgebraicity due to ChanMok (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 crosssection 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 rdimensional affine space in G, which is a crosssection of all regular conjugacy classes of G. In this talk, we will consider natural analogues of Steinberg’s crosssections in the context of a loop group equipped with a Frobenius automorphism. We will show how Steinberg’s crosssection intersects Frobenius twisted conjugacy classes (which are parameterized by "Newton polygons"). Some interesting applications will also be discussed. 
Ye Tian : On Goldfeld conjecture We introduce some recent progress on distribution of 2Selmer groups and Goldfeld conjecture for quadratic twists of elliptic curves. 
Yichao Tian : An prismaticetale comparison theorem in the semistable case Various padic comparison theorems are important topics in padic Hodge theory. In recent years, the prismatic cohomology theorem introduced by Bhatt and Scholze provide us with a uniform framework to compare various padic cohomology theories. In this talk, I will explain a padic comparison theorem between the prismatic cohomology for Fcrystals and the etale cohomology for local systems on semistable padic formal schemes over the ring of integers of a padic fields. 
Liang Xiao : Slopes of modular forms and ghost conjecture The padic valuations of the Up eigenvalues of modular forms are called the (padic) 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 socalled ghost conjecture of BergdallPollack, under certain genericity condition. As a corollary, we resolve many of these conjecutres under the same hypothesis. 
Xinyi Yuan : Geometric BombieriLang conjecture for finite covers of abelian varieties The BombieriLang conjecture is a highdimensional 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 BombieriLang 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 : HarrisVenkatesh 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 Lfunctions and padic Lfunctions at s=0 and s=1 in terms of units and class numbers. The HarrisVenkatesh 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 PrasannaVenkatesh conjectures. In this talk, I will give a picture, formulate a unified conjecture combining HarrisVenkatesh and Stark for weight one modular forms, and describe the proof of this in the imaginary dihedral case. 
Wei Zhang : padic Heights of the arithmetic diagonal cycles on unitary Shimura varieties. We formulate a padic analogue of the Arithmetic GanGrossPrasad Conjectures for unitary groups, relating the padic height pairing of the arithmetic diagonal cycles to the first central derivative (along the cyclotomic direction) of a padic Rankin—Selberg Lfunction 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 padic version of the BlochKato conjecture. This is a joint work with Daniel Disegni. 
Zhiyu Zhang : On Arithmetic Fundamental Lemmas and Arithmetic Transfers The celebrated GrossZagierZhang 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 GanGrossPrasad (AGGP) conjecture relates heights of diagonal cycles on unitary Shimura varieties to central derivatives of automorphic L functions, which has applications to the BlochKato 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 padic 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 BlochKato conjectures. 
