2023
March 31 (Fri) 2023, 10:00 ~ 11:20, 아산이학관 526
Title: Selmer groups of modular forms and modular symbols
Speaker: Chan-Ho Kim (KIAS)
Abstract:
We describe how the structure of Selmer groups of central critical twists of modular forms can be described in terms of modular symbols explicitly. This structural result sometimes reveals more information than the Tamagawa number conjecture of Bloch-Kato.
March 31 (Fri) 2023, 11:20 ~ 12:40, 아산이학관 526
Title: The distribution of the cockernel of a polynomial of a random p-adic matrix
Speaker: Myungjun Yu (Yonsei University)
Abstract:
A random matrix plays an important role in number theory. For example, the distribution of zeros of the Riemann zeta function (of some family of L-functions) is the same as the distribution of eigenvalues of certain random matrices. Also, Friedman--Washington computed the distribution of cockerel of a random p-adic matrix and showed it is the same as Cohen-Lenstra distribution. Recently, Wood generalized the work of Friedman--Washington considering a much more general class of measure on p-adic matrices. In this talk, we discuss the distribution of the cokernel of a polynomial of a random p-adic matrix. This is joint work with Gilyoung Cheong.
April 7 (Fri) 2023, 10:00~11:00
Title: Rational cuspidal subgroup of J_0(N), 아산이학관 526
Speaker: Hwajong Yoo (Seoul National University)
Abstract: For a positive integer N, let X_0(N) be the modular curve over Q associated to the congruent subgroup \Gamma_0(N). Also, let J_0(N) be the Jacobian variety of this modular curve. We discuss a recent progress on the conjecture that the rational cuspidal subgroup of J_0(N) is equal to the rational cuspidal divisor class group of X_0(N).
April 7 (Fri) 2023, 11:30~12:30
Title: 이와사와의 뮤-불변량과 부정방정식 응용, 아산이학관 526
Speaker: Dohyeong Kim (Seoul National University)
Abstract: 수체의 클래스군으로부터 얻어지는 모듈의 대수적 구조로 결정되는 뮤-불변량에 대해 소개하고, 한편으로는 코츠-수자타에 의한 확장을 다루고 또 다른 한편으로는 뮤-불변량의 소멸에 관해 알려진 사실들을 복기한다. 이를 바탕으로 갈루아 코호몰로지군의 소멸에 대한 응용을 얻고자 하며, 이러한 종류의 소멸성이 비가환 샤보티방법론에서 도출되는 부정방정식 해법에 대해 어떤 시사점이 있는지 알아본다.
May 1 (Mon) 2023, 13:00~15:00
Title: Introductory lecture on modular forms, 아산이학관 641-A
Speaker: Subong Lim (Sungkyunkwan University )
Abstract: A modular form plays important roles in number theory. One of famous examples about the importance of modular forms is the proof of the Fermat last theorem by W. Wiles. Moverover, a modular form is related to various areas in mathematics such as number theory, combinatorics, mathematical physics, topology and so on. This lecture introduces basics notions of modular forms and gives theirs basic properties:
1) the definition of slash operator
2) cocyle property of an automorphic factor
3) Fourier exansion of a modular form at a cusp
4) the definition of Hecke operators and their basic properties
May 26 (Fri) 2023, 10:00~11:20
Title: Introduction to arithmetic statistics, 아산이학관 526
Speaker: Jungin Lee (KIAS)
Abstract: Arithmetic statistics is a new branch of mathematics where one studies statistical properties of arithmetic objects in families. In this talk, we introduce the following three topics in arithmetic statistics.
(1) Asymptotic behavior of the number of isogeny/isomorphism classes of abelian varieties over finite fields
(2) Counting number fields by discriminant and counting algebraic tori over Q by Artin conductor
(3) Distribution of random p-adic matrices and random groups
May 26 (Fri) 2023, 11:30~12:50
Title: The average residue of the Dedekind zeta functions associated with non-abelian cubic fields, 아산이학관 526
Speaker: Jaehyun Cho(Peter) (UNIST)
Abstract:
Using the recent result of Bhargava, Taniguchi, and Thorne regarding cubic field count, we compute the average residue of the Dedekind zeta functions associated with non-abelian cubic fields.
June 22 ~ 23 (Fri) 2023
3rd exchange of young ideas in arithmetic (link)
September 15 (Fri) 2023, 10:20~11:20
Title: Core partition and numerical semigroup 아산이학관 526
Speaker: Jaebum Sohn (Yonsei University)
Abstract: In this talk, we'll introduce the t-core partition. The generating function and modularity will be mentioned along with some results and applications of the t-core partition. Recent results on simultaneous core partitions will also be presented. At the end of the talk, we'll introduce numerical semigroup and give some connections between numerical semigroup (numerical sets) and partitions. Some problems related to these topics will be given.
September 15 (Fri) 2023, 11:40~12:40
Title: Distribution of even and odd integers in gaps of numerical semigroups 아산이학관 526
Speaker: Hayan Nam (Duksung Women’s University)
Abstract: A numerical semigroup is a collection of nonnegative integers that includes zero, is closed under addition, and has a finite complement. The gap of a numerical semigroup is defined as the complement of the semigroup. In this talk, we observe the distribution of even and odd integers within the gaps of numerical semigroups.
October 13 (Fri) 2023, 10:20~11:20
Title: Iterated stable number and amicable pairs derived from restricted divisor functions and matrix 아산이학관 526
Speaker: Daeyeoul Kim (Jeonbuk National University)
Abstract: The results of this talk are elementary. It can be seen that various values can be obtained by iterating the restricted divisor functions and matrix. If we consider iteration of t-free divisor functions, then the values of them sees four cases, that is, stable, amicable pair, sociable, unpredictable sequences.
October 13 (Fri) 2023, 11:40~12:40
Title: Quadratic forms with a strong regularity 아산이학관 526
Speaker: Kyoungmin Kim (Hannam University)
Abstract: A (positive definite and non-classic integral) quadratic form is called strongly s-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this talk, we show that for any integer k >= 2, there are only finitely many isometry classes of strongly s-regular quadratic forms with rank k if the minimum of the nonzero squares that are represented by them is fixed.