**Title: **Arithmetic of p-adic Mal'cev-Neumann field

**Abstract: **In this talk, I will briefly introduce the concept of the p-adic Mal'cev-Neumann field and its arithmetic properties. If time permits, I will give some applications of this field in p-adic Hodge theory. This is a joint work with Shanwen Wang.

* We meet on WEDNESDAY.

**Note:** The lunch gathering starts at noon, and the talk begins at 12:15.

**About the Seminar: **This is an in-person seminar at BIMSA over lunch, aimed to promote communications in the Number Theory teams at BIMSA and YMSC. Each talk is 45 minutes long and does not focus on research results. Instead, we encourage each speaker to discuss either (1) a basic notion in Number Theory or related fields or (2) applications or computational aspects of Number Theory. People interested in Number Theory are welcome to attend.

**Upcoming talk:**

The BIMSA-YMSC Number Theory Lunch Seminar meets on Thursdays in Fall 2024.

**Past talks：**

Date: October 16 (Different time!)

Speaker: Yijun Yuan

Title: Arithmetic of p-adic Mal'cev-Neumann field

Abstract: In this talk, I will briefly introduce the concept of the p-adic Mal'cev-Neumann field and its arithmetic properties. If time permits, I will give some applications of this field in p-adic Hodge theory. This is a joint work with Shanwen Wang.

Date: October 10

Speaker: Koji Shimizu

Title: The Chabauty—Coleman method

Abstract: The Chabauty—Coleman method is a p-adic method to determine the rational points on a higher-genus curve. I will apply this technique to prove a theorem of Hirakawa and Matsumura on rational triangles.

Date: September 26

Speaker: Roy Zhao

Title: The Pila-Zannier Method

Abstract: In 2011, Pila proposed a new method using o-minimality and mathematical logic to prove the Andre-Oort Conjecture for modular curves. This framework has been used to great success in unlikely intersection problems, culminating in the proof of the full Andre-Oort Conjecture in 2021. We will demonstrate the method in a simple example and give some hints for how this generalizes.

Date: September 19

Speaker: Yong Suk Moon

Title: Perfectoid spaces in Lean

Abstract: We will briefly discuss how the notion of perfectoid spaces is formalized in Lean.

We met on Thursdays in the Spring of 2024.

Date: May 23

Speaker: Farahnaz Amiri

Title: Calculating the Average Class Numbers of Real Quadratic Fields

Abstract: In this presentation, we will delve into Davenport’s groundbreaking work on calculating the average class number for all real quadratic fields. The concept of class numbers plays a crucial role in algebraic number theory, particularly in the study of quadratic fields. By examining the distribution of class numbers across real quadratic fields, Davenport’s research sheds light on the underlying patterns and properties.

Date: May 16

Speaker: Dong Yan

Title: Eisenstein ideals and Iwasawa invariants

Abstract: We introduce a relation between the generator of the Eisenstein ideal in a Hida family and Iwasawa invariants for modular forms and ideal class groups. We introduce our result focusing on two types of examples.

Date: May 9

Speaker: Dongming She

Title: Local trace formula and branching problems

Abstract: Branching problems study representations of reductive groups over local or global fields which are distinguished by certain nice subgroups. They play important roles in representation theory, arithmetic geometry, and the Langlands program. Many such problems over local fields are proved by purely local techniques using the local trace formula. We will introduce the main ingredients of the local trace formula under this setting, and discuss some of its recent applications.

Date: April 25

Speaker: Heng Du

Title: What is a Bruhat-Tits Building?

Abstract: The theory of Bruhat-Tits buildings was developed to study the structure of reductive groups over local fields. Today, it finds applications across a broad spectrum of mathematical disciplines. In this talk, I will explain the construction of Bruhat-Tits buildings associated with general reductive groups over local fields.

Date: April 18

Speaker: Arnaud Plessis

Title: Why do we construct heights?

Abstract: Heights may be viewed as a tool that turns geometry into arithmetic. I will explain this bridge through a few concrete examples.

Date: March 28

Speaker: Yong Suk Moon

Title: Schemes in Lean

Abstract: We will discuss how the notion of "schemes" in algebraic geometry can be formalized in the Lean theorem prover.

Date: March 21

Speaker: Emmanuel Lecouturier

Title: Modular symbols

Abstract: We will explain the basic theory of modular symbols, e.g. Manin symbols, intersection duality. If time permits, we will give some arithmetic applications.

Date: March 7

Speaker: Koji Shimizu

Title: The p-curvature

Abstract: The p-curvature is a fundamental invariant for connections in characteristic p. I will explain the definition and basic properties.

We met on Wednesdays in the Fall of 2023.

Date: November 29

Speaker: Krishnarjun Krishnamoorthy

Title: Siegel zeros and their consequences in number theory

Abstract: We shall discuss two consequences of the existence of Siegel zeros to arithmetic. The first is regarding class numbers of quadratic fields and here the existence of Siegel zeros is an impediment to proving optimal results. On the other hand, the second is a rather surprising theorem of Heath-Brown which says that the existence of Siegel zeros implies the twin prime conjecture. We shall be brief with proofs and often skip the gory technical lemmas. If we have time, we shall discuss the Deuring-Heilbronn zero repulsion phenomenon.

Date: November 22

Speaker: Lynn Heller

Title: Multiple Zeta Values in the theory of harmonics maps

Abstract: I would like to explain how multiple zeta values appear very naturally when trying to explicitly parametrise very symmetric harmonic maps from surfaces into the round 3-sphere or to the hyperbolic 3-space.

Date: November 15

Speaker: Emmanuel Lecouturier

Title: Half-integral weight modular forms

Abstract: We shall explain some basic facts regarding half-integral weight modular forms, such as the Shimura lift, and give some applications.

Date: November 8

Speaker: Yilong Wang

Title: Numbers in modular tensor categories

Abstract: Modular tensor categories (MTCs) are finite tensor categories giving interesting SL(2, Z) representations. They can be viewed as generalizations of quadratic forms and their Weil representations. In this talk, I will give a quick introduction to MTCs, and briefly explain how, via the congruence property and Galois symmetry of the associated SL(2, Z) representations, basic algebraic number theory appears in the study of MTCs.

Date: November 1

Speaker: Qijun Yan

Title: Uniformization of abelian varieties

Abstract: I will recall the classical uniformization theorem for abelian varieties and introduce its p-adic analogue a la Iovita, Morrow, Zaharescu.

Date: October 25

Speaker: Chuangqiang Hu

Title: Application of Drinfeld modular curves in Coding Theory

Abstract: By Goppa's construction, good towers yield good linear error-correcting codes. The existence of long linear codes with the relative good parameters above the well-known Gilbert-Varshamov bound discovery by Tsfasman et al, provided a vital link between Ihara's quantity and the realm of coding theory. Good towers that are recursive play important roles in the studies of Ihara's quantity, usually constructed from modules curves. Elkies deduced explicit equations of rank-2 Drinfeld modular curves which coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth. In 2015, Bassa, Beelen, Garcia, and Stichtenoth constructed a celebrated (recursive and good) tower (BBGS-tower for short) of curves and outlined a modular interpretation of the defining equations. Soon after that, Gekeler studied in depth the modular curves coming from sparse Drinfeld modules. In this talk, to establish a link between these existing results, I propose a generalized Elkies' Theorem which tells in detail how to directly describe a modular interpretation of the equations of the BBGS tower.

Date: October 18

Speaker: Farahnaz Amiri

Title: The Gauss' class number problems

Abstract: In this talk, first, we will review the problem proposed by Gauss, and their consequences. After that, we talk about the methodology for solving some of these problems.

Date: October 11

Speaker: Dongsheng Wu

Title: The twin prime conjecture and Zhang's work on it

Abstract: I will discuss briefly the history of the twin prime conjecture. In particular, I will explain the celebrated work of Zhang Yitang in attacking this problem.

Date: September 27

Speaker: Tinhinane Amina Azzouz

Title: p-adic differential equations and radii of convergence

Abstract: In the ultrametric setting, linear differential equations present features that do not appear over the complex field. In this talk, I will present some examples of these features. Especially I will talk about the radii of convergence of the solutions, and explain why they are powerful invariants of the equation.

Date: September 20

Speaker: Koji Shimizu

Title: Perfectoid fields and perfectoid spaces

Abstract: In the late 1970s, Fontaine and Wintenberger found that certain interesting fields of characteristic zero and p have the same Galois groups, which eventually led to the theory of perfectoid fields and perfectoid spaces by Scholze. I will explain examples of these mathematical objects and why they are interesting. This is a less technical talk aimed at people in Number Theory and related fields.

Date: September 13

Speaker: No Speaker

Title: Icebreaking and Organization

Abstract: This week, we will introduce ourselves to each other, discuss the organization of the seminar, and exchange the NT seminar and course information in the Beijing area. The lunch talk starts next week.

We met on Thursdays in the Spring of 2023.

Date: June 1

Title: Integer points on a cubic surface and semi-flat Calabi–Yau metrics on special Lagrangian torus bundles

Speaker: Sebastian Heller

Date: May 18

Title: The spectral theory in the sense of Berkovich

Speaker: Tinhinane Amina Azzouz

Date: April 20

Title: Hida families and ideal class groups

Speaker: Dong Yan

Date: April 6

Title: Isogeny graph of supersingular elliptic curves and its applications on cryptography

Speaker: Zheng Xu

Date: March 23

Title: Basic examples of Lean

Speaker: Yong Suk Moon

Date: March 16

Title: Computational thinking and mathematical thinking: a more than beneficial relationship

Speaker: Taiwang Deng

Date: March 2

Title: MW-cohomology and computation

Speaker: Koji Shimizu

We met on Fridays in the Fall of 2022.

Date: November 18

Title: On algebraic Maass forms

Speaker: Taiwang Deng

Date: November 11

Title: On special values of L-functions

Speaker: Emmanuel Lecouturier

Date: November 4

Title: The mu-invariants for residually reducible ordinary Galois representations

Speaker: Dong Yan

Date: October 28

Title: Finite morphism and radii of convergence of a p-adic differential equation

Speaker: Tinhinane Amina Azzouz

Date: October 21

Title: Closed geodesics, class number, and Poincare sections

Speaker: Yitwah Cheung

Date: October 14

Title: Quadratic forms and Bhargava’s compositon law

Speaker: Yong Suk Moon

Date: September 30

Title: A spectral interpretation of the Riemann zeta function

Speaker: Dongsheng Wu

Date: September 23

Title: Drinfeld halfplane

Speaker: Koji Shimizu