초록은 발표 순서대로 쓰여져 있습니다.
Speaker : 이완
Abstract : In their work on l-adic zeta functions and etale cohomology of certain representations in 1973, Coates-Lichtenbaum predicted the exact order of zeros of characteristic polynomials of given Iwasawa modules (of ideal class groups) over the cyclotomic Zp-extension of a given CM field. In this talk, we discuss how we can replace ideal class groups by general class groups. The conjecture for general class groups turns out to be equivalent to two conjectures of Leopoldt and Coates-Lichtenbaum. Moreover, when a given CM field is Galois, this conjecture is reduced to a single statement. This is joint work with Myungjun Yu.
Speaker : 유진주
Abstract : We compute asymptotic formula for the divisor class numbers of cubic Kummer function fields. For computation, we find the mean value of $|L(s, \chi)|^2$ evaluated at $s = 1$ when $\chi$ goes through the primitive cubic Dirichlet characters of $\mathbb{F}_q[T]$, where $L(s, \chi)$ is the associated Dirichlet $L$-function. This is joint work with Jungyun Lee and Yoonjin Lee.
Speaker : 권영욱
Abstract : In 2013, Choi and Kim constructed the canonical basis for the space of weakly holomorphic modular forms for $\Gamma_{0}^{+}(5)$. Later, Kuga proved that the zeros of the canonical basis elements lie in the lower boundary arcs of a fundamental domain. In this talk, we determine the zeros which are algebraic. This is joint work with SoYoung Choi.
Speaker : 박준영
Abstract : In this talk, "genus" means a ring homomorphism from various cobordism rings. I will briefly explain the basic theory of genera, and the relation with modular forms.
Speaker : 박다윤
Abstract : The $m$-gonal number $$P_m(x) = \frac{m-2}{2}(x^2-x)+x$$ where $x \in \mathbb{N}$ which is defined as the total number of dots to constitute the $m$-gon (with $x$ dots for each side) has been popular subject from 2nd Century in B.C in the research of number theory. As one of the most famous stories, Fermat claimed that every positive integer is written as at most $m$ $m$-gonal numbers. Lagrange and Gauss resolved his claim for $m = 4$ and $m = 3$, respectively and finally, Cauchy completed its proof for all $m \geq 3$.
We call a weighted sum of $m$-gonal numbers
$$F_m(\mathbf{x}) = a_1P_m(x_1) + \cdots + a_nP_m(x_n) =: \langle a_1, \cdots, a_n\rangle_m$$
where $a_i \in \mathbb{N}$ an $m$-gonal form. We say that $F_m(\mathbf{x})$ represents $N$ if the diophantine equation $F_m(\mathbf{x}) = N$ has an integer solution $\mathbf{x} \in \mathbb{Z}^n$ for some $N \in \mathbb{N}$.
There are two most typical research items on representation of $m$-gonal forms which are (1) 'universal form' and (2) 'regular form'.
(1) If $F_m(\mathbf{x})$ represents every positive integer, then we say that $F_m(\mathbf{x})$ is universal.
(2) When $F_m(\mathbf{x}) \equiv N \pmod r$ has an integer solution $\mathbf{x} \in \mathbb{Z}^n$ for every modulo $r \in \mathbb{Z}$, we say that $F_m(\mathbf{x})$ locally represents $N$. Super clearly, repersentability implies locally representability, but the converse does not hold in general. We say that $F_m(\mathbf{x})$ is regular if the converse also holds, i.e., if $F_m(\mathbf{x})$ represents every positive integer $N$ which is locally represented by itself.
In this talk, we will see some recent results on representation of $m$-gonal forms. And I would like to suggest highly realizable research project.