**Note:Location & Time can change depending on the speaker's availability.**

This is a research seminar on topics related to number theory and its applications which broadly can include related areas of interests such as analytic and algebraic number theory, algebra, combinatorics, algebraic and arithmetic geometry, cryptography, representation theory etc. The speakers are also encouraged to make their talk more accessible for graduate level students.

For more information,please refer to: http://www.bimsa.cn/newsinfo/647938.html.

### Upcoming Talks:

Speaker: Giada Grossi (Université Sorbonne Paris Nord, CNRS).

Time: 10:00-11:00 Beijing time, April 22, 2024

Title: Kolyvagin’s conjecture and Iwasawa theory

Abstract: Let E be a rational elliptic curve and p be an odd prime of good ordinary reduction for E. In 1991 Kolyvagin conjectured that the system of cohomology classes derived from Heegner points on the p-adic Tate module of E over an imaginary quadratic field K is non-trivial. I will talk about joint work with A.Burungale, F.Castella, and C.Skinner, where we prove Kolyvagin's conjecture in the cases where an anticyclotomic Iwasawa Main Conjecture for E/K is known. Moreover, our methods also yield a proof of a refinement of Kolyvagin's conjecture expressing the divisibility index of the Heegner point Kolyvagin system in terms of the Tamagawa numbers of E.

Location: Shuangqing Comprehensive Building Room B627 (校外双清苑西侧双清综合楼，入口在双清园食堂对面）

### Past Talks:

Speaker: Taiwang Deng (BIMSA)

Time: 10:00-11:00 Beijing time, April 15, 2024

Title: On basic functions for GL_n

Abstract: Let G be a reductive group over a local field F. The basic functions are certain spherical functions defined on G(F), whose Satake transform gives rise to the unramified L factors for an algebraic representation $\rho$ of the Langlands dual group ${^L}G$. They play important roles in the integral representation of global L functions. In general, the structure of such functions is not well understood. In this talk, I will explain how to represent such functions as an integration of certain characteristic functions over the L-monoid for GL_n.

Location: Shuangqing Comprehensive Building Room B627 (校外双清苑西侧双清综合楼，入口在双清园食堂对面）

Speaker: Daxin Xu (MCM)

Time: 10:00-11:00 Beijing time, April 8, 2024

Title: Drinfeld's Lemma for F-isocrystals

Abstract: Drinfeld's lemma for l-adic local systems is a fundamental result in arithmetic geometry. It plays an important role in the Langlands correspondence for a reductive group over the function field of a curve over a finite field, pioneered by Drinfeld for GL_2 and subsequently extended by L. Lafforgue and then V. Lafforgue. In this talk, we will discuss Drinfeld's lemma for p-adic local systems: overconvergent/convergent F-isocrystals. This is based on a joint work with Kiran Kedlaya.

Location: Shuangqing Comprehensive Building Room B627 (校外双清苑西侧双清综合楼，入口在双清园食堂对面）

Speaker: Ruiqi Bai (PKU)

Time: 10:00-11:00 Beijing time, April 1, 2024

Title: Arithmetic Level Raising for U(2r, 1)

Abstract: In this talk, we study the special fiber of U(2r, 1) Shimura varieties at an inert prime. We exhibit elements in the higher Chow group of the supersingular locus and use this to prove the surjectivity of the arithmetic level-raising map. A key ingredient of the proof is to show a form of Ihara’s lemma for odd definite unitary groups. The proof relies heavily on the description of supersingular locus and EKOR stratification on the special fibers with parahoric levels. This is a joint work with Hao Fu. It is inspired by the work of Rong Zhou on quaternionic Shimura varieties, and it can be viewed as an even-dimensional analogue of the ongoing work of Yifeng Liu, Yichao Tian, and Liang Xiao.

Location: Shuangqing Comprehensive Building Room B627 (校外双清苑西侧双清综合楼，入口在双清园食堂对面）

Speaker: Abhinandan (University of Tokyo)

Time: 10:00-11:00 Beijing time, Mar. 25, 2024

Title: Crystalline representations, Wach modules and Prismatic F-crystals

Abstract: For an absolutely unramified extension K/Q_p with perfect residue field, by the works of Fontaine, Colmez, Wach, and Berger, it is well known that the category of Wach modules over a certain integral period ring is equivalent to the category of lattices inside crystalline representations of G_K (the absolute Galois group of K). Moreover, by the recent works of Bhatt and Scholze, we also know that lattices inside crystalline representations of G_K are equivalent to the category of prismatic F-crystals on the absolute prismatic site of the ring of integers of K. The goal of this talk is to present a generalisation of these results to a "small" relative base ring and discuss a direct construction of the categorical equivalence between relative Wach modules and prismatic F-crystals over the absolute prismatic site of the base ring. If time permits, we will also mention relationships between relative Wach modules, q-connections and filtered phi-modules with connections.

This will be an online talk.

Zoom Meeting ID: 4552601552 Passcode: YMSC

https://zoom.us/j/4552601552?pwd=cWxBUjlIN3dxclgrZWFEOC9jcmlwUT09

Title: On stacks of p-adic local shtukas

Speaker: Ian Gleason (Bonn)

Time: 10:00-11:00 Beijing time, Mar 18, 2024

Location: Shuangqing Comprehensive Building Room B627 (校外双清苑西侧双清综合楼，入口在双清园食堂对面）

Abstract:

A classical theorem of Riemann states that complex abelian varieties are classified by their singular homology together with the Hodge filtration. In 2012 Scholze and Weinstein propose a classification of $p$-divisible groups over the $p$-adic complex numbers using analogous linear algebraic data that we may call $p$-adic local shtukas. This development lead to the introduction and study of moduli spaces of $p$-adic local shtukas, which are shown in the Berkeley notes to be generalizations of Rapoport--Zink spaces. In this talk we discuss the v-stack of $p$-adic local and its relation to the moduli problem introduced in the Berkeley notes. We will also discuss two related theorems, the first one explains the relation between the $p$-adic local shtukas and BKF-modules in terms of sheafification. The second one states that stacks of $p$-adic local shtukas are Artin v-stack. The proof of both theorems rely on the theory of kimberlites, we will give an introduction to this theory in the form of a mini-course.

Title: Asai L-functions and arithmetic fundamental lemmas

Speaker: Zhiyu Zhang (Stanford)

Time: 10:00-11:00 Beijing time, Feb 27, 2024

Location: Shuangqing Comprehensive Building Room B627 (校外双清苑西侧双清综合楼，入口在双清园食堂对面）

Abstract:

Asai L-functions for GLn are related to the arithmetic of Asai motives. The twisted Gan—Gross—Prasad conjecture opens a way of studying (a twist of) central Asai L-values via descents and period integrals. I will consider an arithmetic analog of the conjecture on central derivatives using unitary Shimura varieties. I will formulate and prove a twisted arithmetic fundamental lemma.

Title: On exotic Hecke correspondences

Speaker: Pol van Hoften (VU Amsterdam)

Time: 10:00-11:00 Beijing time, January 15th, 2024

Location: Shuangqing Comprehensive Building Room C654 (校外双清苑西侧双清综合楼，入口在双清园食堂对面）

Abstract:

The goal of this talk is to explain joint work in progress with Jack Sempliner on the construction of "exotic" Hecke correspondences between the mod p fibers of different Shimura varieties of Hodge type. Our work generalizes forthcoming work of Xiao-Zhu; our results cover the new situation where the groups underlying the two different Shimura varieties are allowed to be to be non-isomorphic at p. As a consequence of our main results, we obtain exotic isomorphisms of Igusa varieties in the style of Caraiani-Tamiozzo which lead to geometric incarnations of the Jacquet--Langlands correspondence.

Title: Gelfand pairs and gamma factors mod ℓ

Speaker: Robin Zhang (MIT)

Time: 16:00-17:00 Beijing time, January 12th, 2024

Location: Tsinghua University, Jingzhai building, Room 105 (YMSC)

Abstract:

I will discuss two uniqueness phenomena in the representation theory of finite and compact groups. First, we have the classical complex theory of Gelfand pairs and its generalizations; this has many applications to number theory and automorphic forms, such as the uniqueness of Whittaker models and the non-vanishing of the central value of triple product L-functions. Second, we have the local converse theorems that say that irreducible representations have unique Rankin–Selberg gamma factors; in fact, the local Langlands correspondence for GL(n) can be uniquely characterized by gamma factors. We will show how both phenomena can be extended to positive characteristic using two techniques involving module projectivity. The second part of this talk is based on joint work with J. Bakeberg, M. Gerbelli-Gauthier, H. Goodson, A. Iyengar, and G. Moss.

Title: Generalised Euler characteristics of Selmer groups for non-CM elliptic curves

Speaker: Yukako Kezuka（IMJ-PRG）

Time: 10:00-11:00 Beijing time, December 25th, 2023

Location: Shuangqing Comprehensive Building Room C654 (校外双清苑西侧双清综合楼，入口在双清园食堂对面）

Abstract:

We consider an elliptic curve E which does not admit complex multiplication, and discuss its arithmetic over the cyclotomic Z_p-extension and more general p-adic Lie extensions of various base fields F. After reviewing some known results over number fields, we shift our focus to the case of global function fields. In this context, we introduce an invariant which is defined without assuming the finiteness of the Selmer group of E over the base field F.

**Title:** Some uniform Weyl subconvex bounds based on distributional Motohashi's formula

**Speaker:** Han Wu (University of Science and Technology of China)

**Time: **10:00-11:00 Beijing time, Jun. 27, 2023

**Zoom ID:** 455 260 1552（PW: YMSC）

**Location**: Jin Chun Yuan West Building, 3rd floor Lecture Hall (清华大学近春园西楼三楼报告厅）

**Abstract:**

Spectral reciprocities are equalities between moments of automorphic $L$-functions in different families. They are powerful tools for the study of the moment problem and the subconvexity problem. The first spectral reciprocity formula is Motohashi's formula, which relates the cubic moment of $L$-functions for $\GL_2$ with the fourth moment of $L$-functions for $\GL_1$. The exploitation of this formula (over $\mathbb{Q}$) has led Conrey-Iwaniec and Petrow-Young to the uniform Weyl bound for all Dirichlet $L$-functions. In this talk, we will briefly present a distributional version of this formula. Based on this version, applications to the uniform Weyl-type subconvex bounds for some $PGL(2)$ $L$-functions, as well as the limitation of the method, will be discussed. These results are joint works with Balkanova-Frolenkov and Ping Xi, respectively. If time permits, we will present a further application to the partition function, joint work with Nicholas Andersen.

This is the first of two related talks. The other one, which deal with the distributional Motohashi's formula in detail, will be given at the department of mathematics.

**Title:** Square roots of symplectic L-functions and Reidemeister torsion

**Speaker:** Amina Abdurrahman (Stonybrook)

**Time: **16:00-17:00 Beijing time, Jun. 15, 2023

**Zoom ID:** 846 1771 9908（PW: 084284）

**Abstract:**

We give a purely topological formula for the square class of the central value of the L-function of a symplectic representation on a curve. We also formulate a topological analogue of the statement, in which the central value of the L-function is replaced by Reidemeister torsion of 3-manifolds. This is related to the theory of epsilon factors in number theory and Meyer's signature formula in topology among other topics. We will present some of these ideas and sketch aspects of the proof. This is joint work with Akshay Venkatesh.

**Title:** Anticyclotomic p-adic L-functions for Rankin-Selberg products

**Speaker:** Yifeng Liu (IASM, Zhejiang University)

**Time: **10:00-11:00 Beijing time, Jun. 12, 2023

**Location: **Jin Chun Yuan West Building, 3rd floor Lecture Hall (近春园西楼三楼报告厅）

**Zoom ID:** 4552601552 **Passcode: **YMSC

**Abstract:**

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.

**Title:** Holonomic D-modules on rigid analytic spaces

**Speaker:** Andreas Bode (University Wuppertal)

**Time: **16:00-17:00 Beijing time, Jun. 05, 2023

**Zoom ID:** 4552601552 **Passcode: **YMSC

**Abstract:**

Ardakov-Wadsley introduced the theory of co-admissible D-cap-modules on rigid analytic spaces in order to provide a geometric way of studying p-adic locally analytic representations in terms of p-adic differential operators. In this talk, we present an analogue of holonomicity in this framework and discuss parts of a corresponding six-functor formalism, in analogy with Caro's work on overholonomic arithmetic D-modules.

**Title:** Categorical local Langlands

**Speaker:** Zhiyou Wu (BICMR)

**Time: **16:00-17:00 Beijing time, May 22, 2023

**Zoom ID:** 4552601552 **Passcode: **YMSC

**Location: **Jin Chun Yuan West Building, 1st floor Seminar Room (近春园西楼一楼第一报告厅）

**Abstract:**

The local Langlands correspondence is a conjectural relation between sets of smooth representations and Langlands parameters, which has recently been upgraded to categorical equivalences by Fargues-Scholze/Xinwen Zhu in two different ways. I will describe mywork in progress on comparing these two categories.

**Title:** Harder-Narasimhan stratification in p-adic Hodge theory

**Speaker:** Miaofen Chen (East China Normal University)

**Time: **10:00-11:00 Beijing time, May 15 2023

**Zoom ID:** 4552601552 **Passcode: **YMSC

**Location: **Jin Chun Yuan West Building, 3rd floor Lecture Hall (近春园西楼三楼报告厅）

**Abstract:**

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.

**Title:** Flach system on quaternionic Hilbert-Blumenthal surfaces and distinguished periods

**Speaker:** Haining Wang (Fudan University)

**Time: **10:00-11:00 Beijing time, May 8, 2023

**Zoom ID:** 4552601552 **Passcode: **YMSC

**Location: **online seminar (this week only)

**Abstract:**

In this talk, we will report some integrality result on the ratio of the quaternionic distinguished period associated to a Hilbert modular form and the quaternionic Petersson norm associated to a modular form. These distinguished periods are closely related to the notion of distinguished representations that play a prominent role in the proof of the Tate conjecture for Hilbert modular surfaces by Langlands-Rapoport-Harder. Our method is based on an Euler system argument initiated by Flach by producing elements in the motivic cohomologies of the quaternionic Hilbert–Blumenthal surfaces with control of their ramification behaviours. We show that these distinguished periods give natural bounds for certain subspaces of the Selmer groups of these quaternionic Hilbert–Blumenthal surfaces. The lengths of these subspaces can be determined by using the Taylor–Wiles method and can be related to the quaternionic Petersson norms of the modular forms.

**Title:** Mazur's principle for GU(1,2)

**Speaker:** Hao Fu (Université de Strasbourg）

**Time: **10:00-11:00 Beijing time, Apr. 24, 2023

**Zoom ID:** 4552601552 **Passcode: **YMSC

https://zoom.us/j/4552601552?pwd=cWxBUjlIN3dxclgrZWFEOC9jcmlwUT09

**Location: **Jin Chun Yuan West Building, 3rd floor Lecture Hall (近春园西楼三楼报告厅）

**Abstract:**

Mazur's principle gives a criterion under which an irreducible mod l Galois representation arising from a modular form of level Np (with p prime to N) can also arise from a modular form of level N. We prove an analogous result showing that a mod l Galois representation arising from a stable cuspidal automorphic representation of the unitary similitude group G=GU(1,2) which is Steinberg at an inert prime p can also arise from an automorphic representation of G that is unramified at p.

**Title:** Fontaine's conjecture for log p-divisible groups

**Speaker:** Shanwen Wang (Renmin University)

**Time: **10:00-11:00 Beijing time, Apr. 17, 2023

**Zoom ID:** 4552601552 **Passcode: **YMSC

**Location: **Jin Chun Yuan West Building, 3rd floor Lecture Hall (近春园西楼三楼报告厅）

**Abstract:**

Let K be a finite extension of Q_p with ring of integer O_K. It is a very classical result due to Fontaine, Laffaille, Breuil and Kisin that the a galois representation of G_K is cristalline with Hodge-Tate weights in [0,1 ] if and only if it arises from a p-divisible group over O_K. In this talk, we will explain its generalization to log p-divisible groups. More precisely, we show that a galois representation of G_K is semi-stable with Hodge-Tate weights in [0,1 ] if and only if it arises from a log p-divisible group. Joint work with A. Bertapelle and H. Zhao.

**Title: **Splitting models and Galois representations

**Speaker: **Xu Shen (Morningside Center of Mathematics)

**Time: **10-11am, April 10, 2023

**Location:** Jin Chun Yuan West Building, 3rd floor Lecture Hall (近春园西楼三楼报告厅）；ZoomMeeting ID: 4552601552 Passcode: YMSC

https://zoom.us/j/4552601552?pwd=cWxBUjlIN3dxclgrZWFEOC9jcmlwUT09

**Abstract: **

We explain a geometric construction on the Pappas-Rapoport Splitting models of Shimura varieties. More precisely, we construct F-zips with additional structure of fixed type on good reductions of splitting models and their toroidal compactifications. Combined with the method of Goldring-Koskivirta on group theoretical Hasse invariants, this leads to a construction of Galois pseudo-representations associated to torsion classes in coherent cohomology in the ramified setting. This is a joint work with Y. Zheng.

**Title:** Anabelian geometry, effective abc inequalities and their applications

**Speaker:** Ivan Fesenko (University of Warwick)

**Time: **10:00-11:00 Beijing time, Mar. 27, 2023

**Zoom ID:** 4552601552 **Passcode: **YMSC

**Location: **Jin Chun Yuan West Building, 3rd floor Lecture Hall (近春园西楼三楼报告厅）

**Abstract:**

I will introduce some key features of anabelian geometry and the IUT theory of Shinichi Mochizuki, as a short version of longer talks.

Then I discuss several effective abc inequalities established in a recently published paper (Explicit estimates in inter-universal Teichmüller theory, by S. Mochizuki, I. Fesenko, Y. Hoshi, A. Minamide, W. Porowski, Kodai Math. J. 45(2022) 175-236) and explain how their applications change Diophantine geometry.

**Title:** On the Harris-Venkatesh conjecture for weight one forms.

**Speaker:** Emmanuel Lecouturier (YMSC & BIMSA)

**Time: **10:00-11:00 Beijing time, Mar. 20, 2023

**Zoom ID:** 4552601552 **Passcode: **YMSC

**Location: **Jin Chun Yuan West Building, 3rd Floor Lecture Hall (近春园西楼三楼报告厅）

**Abstract:**

Venkatesh recently made very general conjectures regarding the relation between derived Hecke operators and a ``hidden'' action of a motivic cohomology group for an adjoint motive. These conjectures are in the setting of the cohomology of arithmetic groups. Venkatesh and Harris made an analogous conjecture in the setting of coherent cohomology in the first non-trivial case: weight one cuspidal eigenforms. This conjecture has been proved in some dihedral cases by Darmon-Harris-Rotger-Venkatesh recently. I found another approach using triple product L-functions. After some introduction on the conjecture, I will try to explain some ideas behind my method.

**Title:** Recent developments in the theory of p-adic differential equations

**Speaker:** Andrea Pulita (Institut Fourier, Université Grenoble Alpes)

**Time: **16:00-17:00 Beijing time, Mar. 13, 2023

**Zoom ID:** 4552601552 **Passcode: **YMSC

**Abstract:**

I will report on some recent developments in the theory of p-adic differential equations. The talk will be an invitation to the theory and I'll try to maintain it accessible to a large audience.

**Title:** Torsion theorem of the zero of a certain Kodaira-Spencer morphism over P^1 removing four points

**Speaker:** Sheng Mao (YMSC, Tsinghua University & BIMSA)

**Time: **10:00-11:00 Beijing time, Feb. 27, 2023

**Zoom ID:** 4552601552 **Passcode: **YMSC

**Location: **Jin Chun Yuan West Building, No. 1 conference room (近春园西楼第一会议室）

**Abstract:**

In this talk, I shall explain a torsion theorem to the effect that the unique zero of the Kodaira-Spencer map attached to a certain quasi-semistable family of complex projective varieties over the complex projective line is the image of a torsion point of an elliptic curve under the natural projection. The proof is a mod $p$ argument and requires a density of one set of primes. There are three essential ingredients in the proof: a solution to the conjecture of Sun-Yang-Zuo, Pink's theorem, and the Higgs periodicity theorem. This is a joint work with Xiaojin Lin and Jianping Wang.

**Title:** Arithmetic holonomy bounds and their applications

**Speaker: **Vesselin Dimitrov (Institute for Advanced Study in Princeton, Georgia Institute of Technology)

**Time: **20:00-21:30 Beijing time, Jan 17, 2023

**Zoom ID:** 293 812 9202 **Passcode: **BIMSA

**Abstract:**

On the heels of the proof of the Unbounded Denominators conjecture (previously presented in this seminar by Yunqing Tang), we discuss an upgraded and refined form of our main technical tool in this area, the "arithmetic holonomicity theorem," of which we will detail a proof based on Bost's slopes method. Our treatment will lead us to a new alternative argument for the unbounded denominators theorem on the Fourier expansions of noncongruence modular forms. We will then conclude by explaining how the same arithmetic holonomicity theorem also leads to a proof of the irrationality of all products of two logarithms $\log(1+1/n)\log(1+1/m)$ for arbitrary integer pairs $(n,m)$ with $|1-m/n| < c$, where $c > 0$ is a positive absolute constant. This is a joint work with Frank Calegari and Yunqing Tang.

**Title:** Finite Euler products and the Riemann Hypothesis

**Speaker: **Steve M. Gonek (University of Rochester)

**Time: **20:00-21:00 Beijing time, Jan 10, 2023

**Venue:** BIMSA 1118

**Zoom ID:** 293 812 9202 **Passcode: **BIMSA

**Abstract:**

We investigate approximations of the Riemann zeta function by truncations of its Dirichlet series and Euler product, and then construct a parameterized family of non-analytic approximations to the zeta function. Apart from a few possible exceptions near the real axis, each function in the family satisfies a Riemann Hypothesis. When the parameter is not too large, the functions have roughly the same number of zeros as the zeta function, their zeros are all simple, and they repel. In fact, if the Riemann hypothesis is true, the zeros of these functions converge to those of the zeta function as the parameter increases, and between zeros of the zeta function the functions in the family tend to twice the zeta function. They may therefore be regarded as models of the Riemann zeta function. The structure of the functions explains the simplicity and repulsion of their zeros when the parameter is small. One might therefore hope to gain insight from them into the mechanism responsible for the corresponding properties of the zeros of the zeta function.

**Title:** Bounds for standard L-functions

**Speaker: **Paul Nelson (Aarhus University)

**Time: **15:30-16:30 Beijing time, Dec 13, 2022

**Venue:** BIMSA 1131

**Zoom ID:** 293 812 9202 **Passcode: **BIMSA

**Abstract:**

We consider the standard L-function attached to a cuspidal automorphic representation of a general linear group. We present a proof of a subconvex bound in the t-aspect. More generally, we address the spectral aspect in the case of uniform parameter growth.

These results are the subject of the third paper linked below, building on the first two.

**Title:** A Hardy space characterization of the zero-free region of the Riemann zeta function

**Speaker: **Dongsheng Wu (BIMSA)

**Time: **16:00-17:00 Beijing time, Dec 06, 2022

**Zoom ID:** 293 812 9202 **Passcode: **BIMSA

**Abstract:**

In this talk, I will first introduce an equivalent statement of the Riemann Hypothesis in the framework of Hardy spaces in right half-planes. Then I will give a characterization of the zero-free region of the Riemann zeta function in this framework. I will explain the proof and discuss some related topics. This talk is based on a joint work with Fei Wei.

**Title:** A proof of Kudla-Rapoport conjecture for Kramer models at ramified primes

**Speaker: **Qiao He (University of Wisconsin-Madison)

**Time: **Tues.,10:30-11:30am Beijing time, Nov 29, 2022

**Venue:** BIMSA 1131

**Zoom ID:** 293 812 9202 **Passcode: **BIMSA

**Abstract:**

In this talk, I will first talk about the Kudla-Rapoport conjecture, which suggests a precise identity between arithmetic intersection numbers of special cycles on Rapoport-Zink space and derived local densities of hermitian forms. Then I will discuss how to modify the original conjecture over ramified primes and how to prove the modified conjecture. On the geometric side, we completely avoid explicit calculation of intersection number and the use of Tate’s conjecture. On the analytic side, the key input is a surprisingly simple formula for derived primitive local density. This talk is based on joint work with Chao Li, Yousheng Shi and Tonghai Yang.

**Title:** Generalized Paley Graphs, Finite Field Hypergeometric Functions and Modular Forms

**Speaker: **Dermot McCarthy (Texas Tech University)

**Time: **10:30-11:30 Beijing time, Nov 22, 2022

**Venue:** BIMSA 1118

**Zoom ID:** 293 812 9202 **Passcode: **BIMSA

**Abstract:**

In 1955, Greenwood and Gleason proved that the two-color diagonal Ramsey number $R(4,4)$ equals 18. Key to their proof was constructing a self-complementary graph of order 17 which does not contain a complete subgraph of order four. This graph is one in the family of graphs now known as Paley graphs. In the 1980s, Evans, Pulham and Sheehan provided a simple closed formula for the number of complete subgraphs of order four of Paley graphs of prime order.

Since then, \emph{generalized Paley graphs} have been introduced. In this talk, we will discuss our recent work on extending the result of Evans, Pulham and Sheahan to generalized Paley graphs, using finite field hypergeometric functions. We also examine connections between our results and both multicolor diagonal Ramsey numbers and Fourier coefficients of modular forms.

This is joint work with Madeline Locus Dawsey (UT Tyler) and Mason Springfield (Texas Tech University).

**Title:** Quantitative weak approximation of rational points on quadrics

**Speaker: **Zhizhong Huang (AMSS)

**Time: **16:00-17:00 Beijing time, Nov 15, 2022

**Venue:** W11, Ningzhai, Tsinghua University

**Zoom ID:** 293 812 9202 **Passcode: **BIMSA

**Abstract:**

The classical Hasse—Minkowski theorem states that rational points on quadrics (if non-empty) satisfy weak approximation. We explain how Heath-Brown’s delta circle method allows to obtain a quantitive and effective version of this theorem, namely counting rational points of bounded height on quadrics satisfying prescribed local conditions with optimal error terms. We then discuss applications in intrinsic Diophantine approximation on quadrics. This is based on joint work in progress with M. Kaesberg, D. Schindler, A. Shut.

**Title:** Equidistribution in Stochastic Dynamical Systems

**Speaker: **Bella Tobin (Oregon State University)

**Time: **10:30-11:30 Beijing time, Nov 08, 2022

**Venue:** BIMSA 1118

**Zoom ID:** 293 812 9202 **Passcode: **BIMSA

**Abstract:**

In arithmetic dynamics, one typically studies the behavior and arithmetic properties of a rational map under iteration. Instead of iterating a single rational map, we will consider a countable family of rational maps, iterated according to some probability measure. We call such a system a stochastic dynamical system. As such a family can be infinite and may not be defined over a single number field, we introduce the concept of a generalized adelic measure, generalizing previous notions introduced by Favre and Rivera-Letelier and Mavraki and Ye. Generalized adelic measures are defined over the measure space of places of an algebraic closure of the rational numbers using the framework established by Allcock and Vaaler. This turns heights from sums into integrals. We prove an equidistribution result for generalized adelic measures, and in turn prove an equidistribution theorem for random backwards orbits for stochastic dynamical systems. This talk will include some background in arithmetic dynamics and will be suitable for graduate students.

**Title:** Slopes of modular form and ghost conjecture

**Speaker: **Bin Zhao (Capital Normal University)

**Time: **16:00-17:00 Beijing time, Nov 1, 2022

**Venue:** BIMSA 1118

**Zoom ID:** 293 812 9202 **Passcode: **BIMSA

**Abstract:**

In 2016, Bergdall and Pollack raised a conjecture towards the computation of the p-adic slopes of Hecke cuspidal eigenforms whose associated p-adic Galois representations satisfy the assumption that their mod p reductions become reducible when restricted to the p-decomposition group. In this talk, I will report the joint work with Ruochuan Liu, Nha Truong and Liang Xiao to prove this conjecture under mild assumptions. I will start with the statement of this conjecture and the intuition behind it. Then I will explain some strategies of our proof. If time permits, I will mention some arithmetic applications of this conjecture.

**Title:** On $G$-isoshtukas over function fields.

**Speaker: **Wansu Kim

**Time:** 15:00-16:00 Beijing time, Oct 25, 2022

**Zoom ID:** 293 812 9202 **Passcode: **BIMSA

**Abstract:**

Let $F$ be a global function field, and let $G$ be a connected reductive group over $F$. In this talk, we will introduce the notion of $G$-isoshtukas, and discuss a classification result analogous to Kottwitz' classification of local and global $B(G)$. If $G=\GL_n$ then $\GL_n$-isoshtukas are nothing but $\varphi$-spaces of rank $n$ (which naturally arise as an isogeny class of rank-$n$ Drinfeld shtukas), and our classification result for $\GL_n$-isoshtukas can be read off from Drinfeld’s classification of $\varphi$-spaces. This is a joint work with Paul Hamacher.

**Title:** Counting polynomials with a prescribed Galois group

**Speaker: **Vlad Matei (Simion Stoilow Institute of Mathematics of the Romanian Academy)

**Time: **15:30-16:30 Beijing time, Oct 18, 2022 （updated）

**Zoom ID:** 293 812 9202 **Passcode: **BIMSA

**Abstract:**

An old problem, dating back to Van der Waerden, asks about counting irreducible polynomials degree $n$ polynomials with coefficients in the box $[-H,H]$ and prescribed Galois group. Van der Waerden was the first to show that $H^n+O(H^{n-\delta})$ have Galois group $S_n$ and he conjectured that the error term can be improved to $o(H^{n-1})$.

Recently, Bhargava almost proved van der Waerden conjecture showing that there are $O(H^{n-1+\varepsilon})$ non $S_n$ extensions, while Chow and Dietmann showed that there are $O(H^{n-1.017})$ non $S_n$, non $A_n$ extensions for $n\geq 3$ and $n\neq 7,8,10$.

In joint work with Lior Bary-Soroker, and Or Ben-Porath we use a result of Hilbert to prove a lower bound for the case of $G=A_n$, and upper and lower bounds for $C_2$ wreath $S_{n/2}$ . The proof for $A_n$ can be viewed, on the geometric side, as constructing a morphism $\varphi$ from $A^{n/2}$ into the variety $z^2=\Delta(f)$ where each $varphi_i$ is a quadratic form. For the upper bound for $C_2$ wreath $S_{n/2}$ we improve on the monic version of Widmer's result on counting polynomials with an imprimitive Galois group. We also pose some open problems/conjectures.

**Title:** Multizeta for function fields

**Speaker: **Dinesh Thakur (Universty of Rochester)

**Time:** 20:00-21:00 Beijing time, Oct 11, 2022

**Zoom ID:** 293 812 9202 **Passcode: **BIMSA

**Abstract:**

We will discuss multizeta values for the function field case, explain various analogies and contrasts with the rational number field case, and discuss recent developments and open questions.

**Title:** The plectic conjecture over local fields

**Speaker: **Siyan Daniel Li-Huerta

**Time:** 10:00-11:00 Beijing time, Sep 27, 2022

**Zoom ID:** 293 812 9202 **Passcode: **BIMSA

**Room:** BIMSA 1118

Affiliation: Harvard University

Host: Hansheng Diao

**Abstract:**

The étale cohomology of varieties over Q enjoys a Galois action. For Hilbert modular varieties, Nekovář-Scholl observed that this Galois action on the level of cohomology extends to a much larger profinite group: the plectic group. Motivated by applications to higher-rank Euler systems, they conjectured that this extension holds even on the level of complexes, as well as for more general Shimura varieties.

We present a proof of the analog of this conjecture for local Shimura varieties. Consequently, we obtain results for the basic locus of global Shimura varieties, after restricting to a decomposition group. The proof crucially uses a mixed-characteristic version of fusion due to Fargues–Scholze.

**Title:** The Tate conjecture over finite fields for varietes with $h^{2, 0}=1$

**Speaker: **Ziquan Yang

**Time:** 10:00-11:00 Beijing time, Sep 20, 2022

**Zoom ID:** 293 812 9202 **Passcode: **BIMSA

**Room:** BIMSA 1118

**Abstract:**

The past decade has witnessed a great advancement on the Tate conjecture for varietes with Hodge number $h^{2, 0}=1$. Charles, Madapusi-Pera and Maulik completely settled the conjecture for K3 surfaces over finite fields, and Moonen proved the Mumford-Tate (and hence also Tate) conjecture for more or less arbitrary $h^{2, 0}=1$ varietes in characteristic $0$.

In this talk, I will explain that the Tate conjecture is true for mod $p$ reductions of complex projective $h^{2, 0}=1$ when $p>>0$, under a mild assumption on moduli. By refining this general result, we prove that in characteristic $p\geq 5$ the BSD conjecture holds true for height $1$ elliptic curve $\mathcal{E}$ over a function field of genus $1$, as long as $\mathcal{E}$ is subject to the generic condition that all singular fibers in its minimal compactification are irreducible. We also prove the Tate conjecture over finite fields for a class of surfaces of general type and a class of Fano varieties. The overall philosophy is that the connection between the Tate conjecture over finite fields and the Lefschetz $(1, 1)$-theorem over $\mathbb{C}$ is very robust for $h^{2, 0}=1$ varietes, and works well beyond the hyperkahler world. This is a joint work with Paul Hamacher and Xiaolei Zhao.

**Title:** Elementary proofs of Zagier's formula for multiple zeta values and its odd variant

**Speaker:** Li Lai (Tsinghua University)

**Time:** 16:00-17:00 Beijing time,Jul 12, 2022(updated)

**Zoom ID:** 361 038 6975 Passcode: BIMSA

**Room: **BIMSA 1110

**Abstract:**

In 2012, Zagier proved a formula which expresses the multiple zeta values

\[ H(a, b)=\zeta(\underbrace{2,2, \ldots, 2}_{a}, 3, \underbrace{2,2, \ldots, 2}_{b}) \]

as explicit $\mathbb{Q}$-linear combinations of products $\pi^{2m}\zeta(2n+1)$ with $2m+2n+1=2a+2b+3$. Recently, Murakami proved an odd variant of Zagier's formula for the multiple $t$-values

\[ T(a, b)=t(\underbrace{2,2, \ldots, 2}_{a}, 3, \underbrace{2,2, \ldots, 2}_{b}). \]

In this talk, we will give new and parallel proofs of these two formulas. Our proofs are elementary in the sense that they only involve the Taylor series of powers of arcsine function and certain trigonometric integrals. Thus, these formulas become more transparent from the view of analysis. This is a joint work with Cezar Lupu and Derek Orr.

**Title: **Spectrum of p-adic differential equations

**Speaker:** Tinhinane Amina Azzouz (BIMSA)

**Time:** 16:00-17:00 Beijing time, Jun 14, 2022

**Zoom ID: **361 038 6975 Passcode: BIMSA

**Room:** BIMSA 1110

**Abstract:** In the ultrametric setting, linear differential equations present phenomena that do not appear over the complex field. Indeed, the solutions of such equations may fail to converge everywhere, even without the presence of poles. This leads to a non-trivial notion of the radius of convergence, and its knowledge permits us to obtain several interesting information about the equation. Notably, it controls the finite dimensionality of the de Rham cohomology. In practice, the radius of convergence is really hard to compute and it represents one of the most complicated features in the theory of p-adic differential equations. The radius of convergence can be expressed as the spectral norm of a specific operator and a natural notion, that refines it, is the entire spectrum of that operator, in the sense of Berkovich.

In our previous works, we introduce this invariant and compute the spectrum of differential equations over a power series field and in the p-adic case with constant coefficients.

In this talk we will discuss our last results about the shape of this spectrum for any linear differential equation, the strong link between the spectrum and all the radii of convergence, notably a decomposition theorem provided by the spectrum.

Title: Reciprocity, non-vanishing, and subconvexity of central L-values

Speaker: Subhajit Jana (MPIM)

Time: 13:30-15:00 Beijing time, May 26, 2022

Zoom ID: 844 745 8596 Passcode: 568789

Abstract: A reciprocity formula usually relates certain moments of two different families of L-functions which apparently have no connections between them. The first such formula was due to Motohashi who related a fourth moment of Riemann zeta values on the central line with a cubic moment of certain automorphic central L-values for GL(2). In this talk, we describe some instances of reciprocity formulas both in low and high rank groups and give certain applications to subconvexity and non-vanishing of central L-values. These are joint works with Nunes and Blomer--Nelson.

Title:Duals of linearized Reed-Solomon codes

Speaker: Xavier Caruso (CNRS, Université de Bordeaux)

Organiser:Emmanuel Lecouturier (BIMSA)

Time: 16:00-17:00 Friday, 2022/1/7

Zoom: 638 227 8222 PW: BIMSA

Abstract:

Errors correcting codes are a basic primitive which provides robust tools against noise in transmission. On the theoretical perspective, they are usually founded on beautiful properties of some mathematical objects. For example, one of the oldest construction of codes is due to Reed and Solomon and takes advantage of the fact the number of roots of a polynomial cannot exceed its degree. During the last decades, new problems in coding theory have emerged (e.g. secure network transmission or distributive storage) and new families of codes have been proposed. In this perspective, Martínez-Peñas has recently introduced a linearized version of Reed-Solomon codes which, roughly speaking, is obtained by replacing classical polynomials by a noncommutative version of them called Ore polynomials.

In this talk, I will revisit Martínez-Peñas' construction and give a new description of the duals of linearized Reed-Solomon codes. This will lead us to explore the fascinating world of noncommutative polynomials and notably develop a theory of residues for rational differential forms in this context.

Title:Explicit realization of elements of the Tate-Shafarevich group constructed from Kolyvagin classes

Speaker: Lazar Radicevic (Maxplanck Institute, Bonn)

Organiser:Emmanuel Lecouturier (BIMSA)

Time: 16:00-17:00 Wednesday, 2021/12/15

Venue: BIMSA 1118

Zoom: 3885289728 PW: BIMSA

Abstract:

We consider the Kolyvagin cohomology classes associated to an elliptic curve E defined over ℚ from a computational point of view. We explain how to go from a model of a class as an element of (E(L)/pE(L))^Gal(L/ℚ), where p is prime and L is a dihedral extension of ℚ of degree 2p, to a geometric model as a genus one curve embedded in ℙ^(p−1). We adapt the existing methods to compute Heegner points to our situation, and explicitly compute them as elements of E(L). Finally, we compute explicit equations for several genus one curves that represent non-trivial elements of the p-torsion part of the Tate-Shafarevich group of E, for p≤11, and hence are counterexamples to the Hasse principle.

Title：A modular construction of unramified p-extensions of $\Q(N^{1/p})$

Speaker：Jacky Lang (Philadelfia)

Organizer: Emmanuel Lecouturier (BIMSA)

Time：9:00-10:00, Nov. 19, 2021

Venue：BIMSA 1118

Zoom ID: 849 963 1368 Password: YMSC

Abstract:

In Mazur's seminal work on the Eisenstein ideal, he showed that when N and p > 3 are primes, there is a weight 2 cusp form of level N congruent to the unique weight 2 Eisenstein series of level N if and only N = 1 mod p. Calegari--Emerton, Merel, Lecouturier, and Wake--Wang-Erickson have work that relates these cuspidal-Eisenstein congruences to the p-part of the class group of $\Q(N^{1/p})$. Calegari observed that when N = -1 mod p, one can use Galois cohomology and some ideas of Wake--Wang-Erickson to show that p divides the class group of $\Q(N^{1/p})$. He asked whether there is a way to directly construct the relevant degree p everywhere unramified extension of $\Q(N^{1/p})$ in this case. After discussing some of this background, I will report of work with Preston Wake in which we give a positive answer to this question using cuspidal-Eisenstein congruences at prime-square level.

Title：The unbounded denominators conjecture

Speaker：Yunqing Tang (Princeton university)

Organizer: Emmanuel Lecouturier (BIMSA)

Time：9:30-10:30, Oct. 29, 2021

Venue：BIMSA 1118

Zoom ID: 849 963 1368 Password: YMSC

Abstract:

The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer, asserts that a modular form for a finite index subgroup of SL_2(Z) whose Fourier coefficients have bounded denominators must be a modular form for some congruence subgroup. In this talk, we will give a sketch of the proof of this conjecture based on a new arithmetic algebraization theorem. (Joint work with Frank Calegari and Vesselin Dimitrov.)

Title：Eisenstein congruences and Euler systems

Speaker：Oscar Rivero Salgado (University of Warwick)

Organizer: Emmanuel Lecouturier (BIMSA)

Time：16:00-17:00, Oct. 22, 2021

Venue：BIMSA 1118

Zoom ID: 388 528 9728 Password: BIMSA

Abstract:

Let f be a cuspidal eigenform of weight two, and let p be a prime at which f is congruent to an Eisenstein series. Beilinson constructed a class arising from the cup-product of two Siegel units and proved a relationship with the first derivative of the L-series of f at the near central point s=0. I will motivate the study of congruences between modular forms at the level of cohomology classes, and will report on a joint work with Victor Rotger where we prove two congruence formulas relating the Beilinson class with the arithmetic of circular units. The proofs make use of Galois properties satisfied by various integral lattices and exploits Perrin-Riou's, Coleman's and Kato's work on the Euler systems of circular units and Beilinson-Kato elements and, most crucially, the work of Fukaya-Kato around Sharifi’s conjectures.

Title: Modular regulator with Rogers-Zudilin method

Speaker: Weijia Wang (ENS Lyon)

Time: 2020-7-14, 16:00 – 17:00

Abstract: Let Y (N) be the modular curve of level N and E(N) be the universal elliptic curve over Y (N). Beilinson (1986) defined the Eisenstein symbol in the motivic cohomology of Ek(N) and the work of Deninger–Scholl (1989) shows the Petersson inner product of its regulator gives us special L-values. In this talk I will present how to relate the modular regulator with L-value of quasi-modular forms by using Lanphier’s formula and Rogers–Zudilin method.

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Title: Projective bundle theorem in MW-motives

Speaker: Nanjun Yang (YMSC, Tsinghua)

Time: 2020-7-2, 10:00 – 11:00

Abstract: We present a version of projective bundle theorem in MW-motives (resp. Chow-Witt rings), which says that $\widetilde{CH}^*(\mathbb{P}(E))$ is determined by $\widetilde{CH}^*(X)$ and $\widetilde{CH}^*(X\times\mathbb{P}^2)$ for smooth quasi-projective schemes $X$ and vector bundles $E$ over $X$ with odd rank. If the rank of $E$ is even, the theorem is still true under a new kind of orientability, which we call it by projective orientability. As an application, we compute the MW-motives of blow-ups over smooth centers. (arXiv 2006.11774)

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Title: Elliptic cocycle for GLN(Z) and Hecke operators

Speaker: Hao Zhang (Sorbonne Université)

Time: 2020-7-2, 16:00 – 17:00

Abstract: A classical result of Eichler, Shimura and Manin asserts that the map that assigns to a cusp form f its period polynomial r_f is a Hecke equivariant map. We propose a generalization of this result to a setting where r_f is replaced by a family of rational function of N variables equipped with the action of GLN(Z). For this purpose, we develop a theory of Hecke operators for the elliptic cocycle recently introduced by Charollois. In particular, when f is an eigenform, the corresponding rational function is also an eigenvector respect to Hecke operator for GLN. Finally, we give some examples for Eisenstein series and the Ramanujan Delta function.

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Meeting ID: 916 5344 6007

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