Number Theory
Abstracts
2월 4일 (일)
Speaker : 오병권
Title : TBA
Abstract : TBA
Speaker : Filip Najman (Talk 1)
Title : Elliptic curves
Abstract : In this talk we introduce some basic notions and results about elliptic curves. We will focus mostly on isogenies of elliptic curves, torsion points and Galois representations attached to elliptic curves.
Speaker : Petar Orlic
Title : Gonality of the modular curve X_0(N)
Abstract : Modular curves are one of the significant objects in number theory. One of their important properties is that points on modular curves represent classes of elliptic curves with some additional structure. Therefore, we can answer some questions regarding elliptic curves by studying modular curves. Gonality of an algebraic curve is defined as the minimal degree of a nonconstant morphism from that curve to the projective space P^1. I will talk about the methods used to determine the Q-gonality of the modular curve X_0(N).
2월 5일 (월)
Speaker : 이계선
Title : Triangle groups in SL(3,Z)
Abstract : In this talk, I will explain how to construct interesting examples of compact hyperbolic triangle groups in SL(3,Z). Joint work with Jaejeong Lee and Florian Stecker.
Speaker : Filip Najman (Talk 2)
Title : Modular curves
Abstract : Modular curves are moduli spaces of elliptic curves whose image of their mod N Galois representaions is contained in some fixed subgroup of $GL_2(\mathbb Z /N \mathbb Z)$, up to conjugacy. We will describe known results and techniques for studying rational and low degree points on modular curves.
Speaker : 김인서
Title : The number of representations of totally positive integers as the form of x^2+y^2+2z^2 in Q(sqrt 5)
Abstract : There has been significant interest in computing the number of ways to represent an integer as the sum of squares. In 2018, Shigeaki Tsuyumine computed the number of representations of totally positive integers in a number field Q(sqrt 5) as sums of three integral squares, from the coefficients of the Shimura lift of the third power of theta series. Based on their method, I will introduce the computation of the number of representations of totally positive integers as the form of x^2+y^2+2z^2, which is a universal form of Q(sqrt 5).
Speaker : 최형민
Title : Bridging Algebraic Number Theory to Post-Quantum Digital Signatures
Abstract : How can we define security? Classical cryptographic analyses hinge on algebraic and number-theoretic hard problems like integer factorization, discrete logarithms over finite fields, or elliptic curves. The security of the schemes can be proven if at least the underlying problems are hard enough. However, with the surge in quantum threat awareness on the hardness of the classical problems, the spotlight has shifted to security against quantum computers and algorithms, a.k.a. classical cryptography to post-quantum cryptography. Lattice-based cryptography is one of the most popular fields in post-quantum cryptography, with a strong security guarantee based on reductions.
This talk centers on introducing HAETAE, a lattice-based digital signature scheme. We begin by introducing the hard lattice problems and the related reductions for security. This exploration lays the groundwork for understanding digital signatures and culminates by introducing HAETAE, a post-quantum digital signature.
2월 6일 (화)
Speaker : 박지훈
Title : Hodge symmetries on the Jacobian rings of smooth projective hypersurfaces
Abstract : Let $F$ be a number field and $n$ be a positive integer. Let $X_G\hookrightarrow \BP^n_F$ be a smooth projective hypersurface defined by a homogeneous polynomial $G(\ud x) \in F[\ud x]$. The primitive middle-dimensional de Rham cohomology $H_{dR,\sigma}:=H_{dR,\pr}^{n-1}(X_G;F) \otimes_{F,\sigma} \C$, where $\sigma:F \hookrightarrow \C$ is a real embedding, has the Hodge decomposition $\bH_0:=\bigoplus_{p+q=n-1}\bH_0^{p,q}$.
The goal of this article is to describe explicitly the involution $c_{dR}$ on $H_{dR,\sigma}$, which corresponds to the complex conjugation on $\bH_0$ switching $\bH_0^{p,q}$ and $\bH_0^{q,p}$ in the Hodge decomposition, through Griffiths' description of $H_{dR,\pr}^{n-1}(X_G;F)$ in terms of the Jacobian ideal of $G(\ud X)$. The talk is based on the joint work with Junyeong Park and Philsang Yoo.
Speaker : 이슬비
Title : Diophantine approximation by rational numbers of certain parity types
Abstract : In the study of Diophantine approximation, a natural question is which rationals p/q minimize |qx-p| with a bounded condition over q. We call such rationals the best approximations. The regular continued fraction provides an algorithm for generating the best approximations. From a broader perspective, we are interested in the best approximations with congruence conditions on their numerators and denominators. It is known that the continued fraction allowing only even integer partial quotients generates the best approximations whose numerator and denominator have different parity. In this talk, we will reconstruct the even continued fraction map using a symbolic sequence of real numbers associated with a specific triangle group in the upper half-plane. Subsequently, we will define interval maps that yield algorithms for generating best approximations with other parity conditions. This is joint work with Dong Han Kim and Lingmin Liao.
Speaker : 이석형
Title : Lifting problem for universal quadratic forms
Abstract : A positive definite quadratic form is called universal if it can represent all nonnegative integers, such as the form $x^2+y^2+z^2+w^2$ as shown by Lagrange. The concept of "lifting problem" generalizes this notion to totally real number fields $K$: an integral quadratic form is called universal over $K$ if it can represent every total positive integral element. In this talk, we survey several results on universal forms over number fields, especially focusing on some recent developments obtained by utilizing the geometry of numbers argument. Then we present our result classifying all cubic and biquadratic fields which admit an integral universal form. Joint work with Daejun Kim (KIAS).