3rd exchange of young ideas in arithmetic
When: June 22 (Thur), 2023 ~ June 23 (Fri), 2023
Where: 고려대 아산이학관 117호
Schedule
6월 22일 (목요일)
12시 40분 ~ 13시 등록
13시 ~ 14시 특강 1 (이윤복 인천대)
14시 20분 ~ 15시 권홍 (UNIST)
15시 10분 ~ 15시 50분 김한솔 (KAIST)
16시 10분 ~ 16시 50분 이승민 (고려대)
17시 10분 ~ 17시 50분 민준휘 (UNIST)
6월 23일 (금요일)
9시 30분 ~ 10시 30분 특강 2 (이윤복 인천대)
10시 50분 ~ 11시 30분 이원웅 (KAIST)
11시 40분 ~ 12시 20분 유재광 (고려대)
Speakers and abstracts
이윤복(인천대)
1. 제목 : Introduction to the theory of the Riemann zeta function
초록 : We introduce basic properties of the Riemann zeta function, focusing on the analytic continuation and the functional equation of the Riemann zeta function.
2. 제목 : Classification of L-functions of degree 2 and conductor 1
초록 : In this talk, we introduce the main theorem in the paper "classification of L-functions of degree 2 and conductor 1" by Kaczorowski and Perelli, Adv. Math. 408 (2022), part A, Paper No. 108569, 46 pp. We provide a sketched proof focusing on an analytic continuation of the standard twist of L-functions in the extended Selberg class. A preprint is available at https://arxiv.org/abs/2009.12329.
권홍 (UNIST)
제목: A relation between the decimal expansion and the continued fraction expansion of a real number
초록: Using some results of ergodic theory, we can prove the Loch's theorem, which gives us information about the relation between the decimal expansion and the continued fraction expansion of a real number. In this talk I will give a sketch of the Loch's theorem and a generalization of it. In addition, I will introduce some result about the distribution of $k_{n}(x)$, which is the agreement of the continued fraction expansions of n'th decimal approximations, i.e, $y_{n}=(\lfloor 10^{n}x \rfloor)/10^{n}$ and $z_{n}=y_{n}+1/10^{n}$.
김한솔 (KAIST)
제목: Density of elliptic curves over number fields with prescribed torsion subgroups
초록: Let $K$ be a number field. For positive integers $m$ and $n$ such that $m \mid n$, we let $\mathscr{S}_{m,n}$ be the set of elliptic curves $E/K$ defined over $K$ such that ${E(K)}_{tors} \supseteq \mathscr{T} \cong \mathbb{Z}/m\mathbb{Z} × \mathbb{Z}/n\mathbb{Z}$. We prove that if the genus of the modular curve $X_{1}(m, n)$ is $0$, then `almost all' $E \in \mathscr{S}_{m,n}$ satisfy that ${E(K)}_{tors} = \mathscr{T}$ , i.e., no larger than $\mathscr{T}$. In particular, if $m = n = 1$, we generalize the results of Duke and Zywina over a number field $K$ such that $K \cap \mathbb{Q}^{cyc} = \mathbb{Q}$ to arbitrary number fields $K$ for the trivial torsion subgroup.
This is a joint work with Bo-Hae Im.
민준휘 (UNIST)
제목: Generation of Hecke fields by products of modular L-values
초록: Let f, g be newforms of weight 2k and level R. For odd prime p, we let \chi be Dirichlet characters of conductor p^h and order p^{h-1}. Due to the work of Shimura, it is known that there exist algebraic modular L-values L_f(\chi) and L_g(\chi). We prove that the Hecke field Q_fQ_g(\zeta_{h-1}) is generated by the products L_f(\chi)L_g(\chi).
유재광 (고려대학교)
제목 : Proportion of modular forms with transcendental zeros for general levels
초록 : Let Γ be a congruence subgroup such that Γ1(N)⊂Γ⊂Γ0(N) for some positive integer N. For a positive integer k, let Mk,Z(Γ) be the set of modular forms of weight k on Γ with integral Fourier coefficients. Let Rk(Γ) be the set of common zeros in the upper half plane H of all the modular forms of weight k on Γ. In this talk, we will introduce studies of zeros of modular forms. In addition, we will discuss that Mk,Z(Γ) has density 1 for modular forms of which every algebraic zero is in Rk(Γ).
이승민 (고려대학교)
제목 : Ihara zeta function for finite graph with circuit rank two
초록 : In this talk, we will introduce the prerequisite knowledge necessary to understand the Ihara zeta function and its determinant formula. We will then define the Ihara zeta function and its determinant formula. Additionally, we will explain the definition of finite graphs with circuit rank two and their values for the Ihara zeta function.
이원웅 (KAIST)
제목 : On the zeros of depth 1 quasimodular forms
초록 : In this talk, we discuss the critical points of modular forms, or more generally the zeros of quasimodular forms of depth 1 for the full modular group. In particular, we consider the derivatives of the unique weight k modular forms with the maximal number of consecutive zero Fourier coefficients following the constant 1. This is joint work with Bo-Hae Im.
조직위원 유재광, 이승민, 최도훈