2023 정기 정수론 학회

초록: One of the central themes in number theory is to understand analytic properties of Riemann-zeta functions. Among them, the infinite product, called “Euler product” seems to be natural, but for the arbitrary infinite sum, one cannot expect this fascinating structure. To resolve the difficulty, Rankin and Selberg construct integrals of a pair of cuspidal holomorphic forms against Eisenstein series. This is known as Rankin-Selberg method. Along the line of the historical development, the purpose of my first talk is devoted to introducing aforementioned objects in the classical language. Continued with the second talk, we will explore, so to speak, “newforms” which are in a special kind of holomorphic forms. After translating them in adelic setting, we precisely “evaluate” integrals. The evaluation is difficult in general, and it is an extreme incidence of more general results of my recent joint work with Peter Humphries. This talk will be accessible to general audiences in graduate levels and delivered in Korean.

초록: Theta series of a (positive-definite and integral) quadratic form Q is the Fourier series generating function for the number of representations of integers by Q. It is well known that this theta series is a modular form. In this talk, we discuss the decomposition of theta series for quadratic forms in three variables with congruence conditions in terms of lattice cosets. We show that the natural decomposition into an Eisenstein series, a unary theta function, and a cuspidal form which is orthogonal to unary theta functions correspond to the theta series for the genus, the deficiency of the theta series for the spinor genus from that of the genus, and the deficiency of the theta series for the class from that of the spinor genus, respectively. This extends known results for lattices. This is joint work with Ben Kane.

초록: A positive-definite and integral quadratic form f is called regular if it represents all integers that are locally represented.

A regular quadratic form f is called new if there is no proper subform g of f which represents all integers represented by f. In this talk, we show that the rank of any new regular quadratic form is bounded by an absolute constant.

This is a joint work with Byeong-Kweon Oh.

초록: Core partitions have played important roles in the theory of partitions and related areas. In this talk, we briefly summarize interesting and important results on t-cores from classical results like how to obtain a generating function to recent results like simultaneous cores. This includes the modularity of t-core partition generating functions, the existence of t-core partitions, asymptotic formulas and arithmetic properties of t-core partitions, and combinatorial and number theoretic aspects of simultaneous core partitions. This talk is based on the article "A survey on t-core partitions" with Byungchan Kim, Hayan Nam, and Jaebum Sohn

초록: One of the central themes in number theory is to understand analytic properties of Riemann-zeta functions. Among them, the infinite product, called “Euler product” seems to be natural, but for the arbitrary infinite sum, one cannot expect this fascinating structure. To resolve the difficulty, Rankin and Selberg construct integrals of a pair of cuspidal holomorphic forms against Eisenstein series. This is known as Rankin-Selberg method. Along the line of the historical development, the purpose of my first talk is devoted to introducing aforementioned objects in the classical language. Continued with the second talk, we will explore, so to speak, “newforms” which are in a special kind of holomorphic forms. After translating them in adelic setting, we precisely “evaluate” integrals. The evaluation is difficult in general, and it is an extreme incidence of more general results of my recent joint work with Peter Humphries. This talk will be accessible to general audiences in graduate levels and delivered in Korean.

초록: Modular forms appear in many ways in number theory and they are presently at the center of an immense amount of research activity. Among them, the set of weakly holomorphic modular forms in known as an infinite dimensional vector space, and although the canonical basis (that is reduced row echelon basis) of the space $M_{k}^{!}(N)$ of weakly holomorphic modular forms has many arithmetical properties, the general method of obtaining the canonical basis of it is still being studied.

In this talk, we discuss the recipe for finding a canonical basis of the space $M_{k}^{!}(N)$ of weakly holomorphic modular forms and its applications.

초록: The language of semi-stable p-adic Galois representations can be translated into that of weakly admissible filtered $(\phi, N)$-modules of finite rank. The advantage of the translation is that one can explicitly calculate such modules. In this talk, we study these phenomenons and calculate the cases of dimension smaller than or equal to 3.

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.