Title: On composition laws
Abstract: In this talk, we will present a short review of Gauss and Bhargava composition laws and their consequences for the class group of real quadratic fields.
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 Spring 2025.
Date: March 6
Speaker: Koji Shimizu
Title: Rational points and fundamental groups
Abstract: I will introduce Grothendieck's Section Conjecture, which describes the rational points of a hyperbolic curve over a number field in terms of its arithmetic etale fundamental group.
Date: March 20
Speaker: Satyabrat Sahoo
Title: On the solutions of S-unit equations and their applications
Abstract: Let F be a number field, and let S be a finite set of prime ideals of F. In this talk, we will discuss the solutions to the S-unit equation x+y=1, where x and y are S-units. Additionally, we will provide various applications of this discussion. For instance, when S consists of all prime ideals of F that lie above 2, we will explore the relationship between the solutions of the S-unit equation and the solutions of the Fermat equation x^p+y^p=z^p over F.
Date: March 27
Speaker: Yongxiong Li
Title: Iwasawa theory and tame kernels
Abstract: We will talk about an early work of Coates, who used the fundamental results of Iwasawa and Tate, gave the asymptotic behaviour of the order of the p-primary part of K_2 of the integer rings along the cyclotomic Z_p extension over number field.
Date: April 3
Speaker: Peter Koroteev
Title: q-Difference Equations and p-Curvature
Abstract: The concept of $p$-curvature originated in Grothendieck's unpublished work from the 1960s and was subsequently developed further by V. Katz. The $p$-curvature plays an important role in the theory of ordinary differential equations (ODEs) as well as holonomic PDEs, establishing a connection between the existence of algebraic fundamental solutions and their behavior under reduction modulo a prime $p$.
In this talk, I will describe how the $p$-curvature can be obtained from a reduction of a quantum difference equation on a Nakajima quiver variety $X$ over the field of finite characteristic. Using a Frobenius map, I will show how to calculate the spectrum of this $p$-curvature.
Date: April 10
Speaker: Dongsheng Wu
Title: Additive Triples of Bijections and the Circle Method on Finite Groups
Abstract: Let G be a finite group of order n, and let S be the set of all bijections from {1, 2,..., n} to G. We say that three bijections \pi_1, \pi_2, \pi_3 in S form an additive triple if \pi_1 + \pi_2 = \pi_3. In this talk, we explore how ideas from analytic number theory—particularly the circle method—can be adapted to estimate the number of such additive triples.
Date: April 24
Speaker: Sergey Oblezin
Title: Archimedean L-functions and Hecke-Baxter operators
Abstract: For an admissible representation of a split real group GL(n,R), one might canonically attach a local Archimedean L-factor, a function in a single complex variable capturing data of the original representation. Introducing this specific L-function is motivated by deep connections with the theory of zeta-functions of global number fields. In particular, the local L-factors of representations of non-compact groups play an essential role in formulation of the Langlands correspondence for these groups. However, the classical construction of the Archimedean L-factors follows some round-about strategy. In my talk I discuss the so-called Hecke-Baxter operator, an element of the spherical Hecke algebra associated with the Gelfand pair (GL(n,R), O(n)). This element is specified by the property that its action on a O(n)-fixed vector in a principal series GL(n,R)-representation reproduces the Archimedean L-function attached to this representation. This is a joint project with A. Gerasimov and D. Lebedev.
Date: May 8
Speaker: Cezar Lupu
Title: Multiple Zeta Values
Abstract: Multiple zeta values (abbreviated as MZV's), or sometimes called Euler-Zagier sums, are important objects at the interface of different fields of mathematics and physics spanning from analysis, number theory, geometry to perturbative quantum field theory. In this talk, we explore this world of MZV's with an emphasis on their analytic aspects such as linear relations and identities involving different families of multiple zeta values. Moreover, we state and give some insights of important conjectures that will give us a better understanding of these linear relations among MZV's.
Date: May 22
Speaker: Farahnaz Amiri
Title: On composition laws
Abstract: In this talk, we will present a short review of Gauss and Bhargava composition laws and their consequences for the class group of real quadratic fields.
Past talks：
We met on Wednesdays in the Fall of 2024.
Date: December 12
Speaker: Nanjun Yang
Title: Witt group of nondyadic curves
Abstract: The Witt group of an algebraic variety is the Grothendieck group of vector bundles with non-dengenerate symmetric inner products modulo those with Langrangians. For a field it classifies quadratic forms and is built by extensions of etale cohomologies. For real varieties, it's known to be related to connected components of real points. For curves over local fields, only the case of hyperelliptic curves was considered by Parimala, Arason et. al..
In this talk, we show that Witt group of smooth projective curves over nondyadic local fields is determined by the Picard group, graph of special fiber, splitness of torus and existence of rational points of odd degrees. We compute the case of elliptic curves as an example.
Date: December 5
Speaker: Krishnarjun Krishnamoorthy
Title: Resolution of cusp singularities and class numbers
Abstract: We will revisit the resolution of cuspidal singularities of Hilbert modular surfaces following Hirschburg. We shall also mention an interesting connection to the special values of Shimizu L functions.
Date: November 21
Speaker: Xiuwu Zhu
Title: Ideal Class Groups of Number Fields in Zp-Extensions
Abstract: Let $F$ be a number field and $F_\infty$ be a $\mathbb{Z}_p$-extension of $F$. Iwasawa established certain patterns in the behavior of the class groups of the intermediate fields in $F_\infty/F$. Subsequently, many researchers have conducted more detailed studies of the class groups for various choices of $F$ and different types of $\mathbb{Z}_p$-extensions. In this talk, we will review known results and outline the proofs for some examples.
Date: November 7
Speaker: Qijun Yan
Title: Introduction to Deligne's Period Conjecture
Abstract: Deligne's period conjecture predicts relations between the critical values of motivic L-functions and certain period matrices. I will introduce this conjecture for algebraic varieties over number fields.
Date: October 31
Speaker: Dali Shen
Title: Interlacing property and the Prime Number Theorem
Abstract: The interlacing property of Beukers-Heckman, which is used to determine the finiteness of the monodromy groups of the higher-order hypergeometric equations, also appears in the proof by Chebyshev of the weak version of the Prime Number Theorem. I will explain this relationship observed by Rodriguez-Villegas.
Date: October 24
Speaker: Dong Yan
Title: Modular forms with complex multiplication
Abstract: In this talk, I will introduce Ribet's paper on modular forms with complex multiplication and its Galois representations.
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
