김대열 (전북대)

제목: Menon-type identities with characters

초록: In this talk, we consider the actions of subgroups of the general linear group $GL_r (\Bbb Z_n )$ on $\Bbb Z_n^r$, including groups of upper triangular matrices in $GL_r (\Bbb Z_n )$, unipotent groups, Heisenberg groups and extended Heisenberg groups. By applying the Cauchy-Frobenius-Burnside lemma, we will state several generalizations of the well known Menon's identity \cite{LK}, \cite{Menon}, \cite{Z-C}.

김경민 (성균관대)

제목: A sum of squares not divisible by a prime.

초록: Let $p$ be a prime. We define $S(p)$ the smallest number $k$ such that

every positive integer is a sum of at most $k$ squares of integers that are not divisible by $p$.

In this article, we prove that $S(2)=10$, $S(3)=6$, $S(5)=5$, and $S(p)=4$ for any prime $p$ greater than $5$.

In particular, it is proved that every positive integer is a sum of at most four squares not divisible by $5$,

except the unique positive integer $79$. This is a joint work with Byeong-Kweon Oh.

송경환 (고려대)

제목: The inverses of tails of the Riemann zeta function for some natural numbers and real numbers in critical strip

초록: In this talk, we give some results regarding the Riemann zeta function and its variations. We begin the talk by introducing (1) The Riemann zeta function and its generalized functions. (2) Some properties of the functions related to the Riemann zeta function. (3) A reciprocal sum related to the Riemann zeta function at s = 2, 3, 4 and 5, as preliminaries. Afterwards, we present a somewhat new result on the reciprocal sum related to the Riemann zeta function at s = 6, which is a joint work with Dr. WonTae Hwang. Also, we give some bounds of the inverses of tails of the Riemann zeta function on 0 < s < 1 and compute the integer parts of the inverses of tails of the Riemann zeta function for s =1/2, 1/3 and 1/4, which is a joint work with Prof. Donggyun Kim.

윤동성 (부산대)

제목: Form class groups for extended ring class fields

초록: Let $K$ be an imaginary quadratic field of discriminant $d_K$ and let $Q(d_K)$ be the set of primitive positive definite binary quadratic forms of discriminant $d_K$. Then, the modular group $\mathrm{SL}_2(\mathbb{Z})$ gives an equivalence relation on $Q(d_K)$ and the set of equivalence classes becomes a group called the form class group by the composition law given by Gauss. This group is isomorphic to the Galois group of the Hilbert class field of $K$. In this talk, we consider the modifications of the set of quadratic forms and the congruence group and define the new group of form classes which is isomorphic to the Galois group of an extended ring class field of $K$.

이철희 (고등과학원)

제목 : Quadratic forms over 2-adic integers with computers

초록 : It is often painful to work with quadratic forms over 2-adic integers. In his book `Rational Quadratic Forms' Cassels wrote ``only the masochist is invited to read the rest of this section (on canonical forms of quadratic forms over 2-adic integers)'' While writing a computer program to compute the Gross-Keating invariant of a quadratic form over p-adic integers, I had to consider various matrix reduction procedures, including the case of $p=2$. I will talk about how computers can help us in this subject.

정근영 (유니스트)

제목 : The root number of Hecke character associated to an elliptic curve with complex multiplication

초록 : Let $E/F$ be an elliptic curve with complex multiplication by an imaginary quadratic field $K$. When $K$ is included in $F$, then various arithmetic invariants (algebraic rank, analytic rank, dimension of Selmer group, root number) of $E$ are always even, under the Birch—Swinnerton-Dyer conjectures. It is considered that those arithmetic invariants are not just even, but a twice of other arithmetic invariants. For example, the root number of an elliptic curve is a square of the root number of the associated Hecke character. We will talk about a problem on distribution of the root numbers of Hecke character, which is a Hecke-character-analogue of the work of D. Byeon and N. Kim. This is a joint work in progress with J. Kim and T. Kim.

조성문 (포스텍)

제목 : On smooth integral models of p-adic reductive groups

초록 : A smooth integral model of p-adic reductive group has an important application to the classification of integral bilinear forms (e.g. quadratic or hermitian forms). It is also crucially used to calculate the dimension of the space of algebraic automorphic forms and the space of modular forms on Shimura varieties.

In this talk, I will explain a conjectural construction of smooth integral models of any p-adic reductive groups.

황원태 (고등과학원)

제목 : Automorphism groups of polarized abelian surfaces over finite fields

초록 : Let X be an abelian surface over a field k. It is known that X is either simple or isogenous to a product of two elliptic curves over k. It is then natural to consider the group Aut_k(X) of automorphisms of X over k. In general, the group Aut_k(X) is not finite. Hence, we might be led to consider the classification of finite subgroups of Aut_k(X) in a suitable sense. In such a classification, the notion of a polarized abelian surface could be more appropriate. In this talk, we briefly give such a classification when the ground field k is a finite field. After introducing several basic facts related to our goal, as our preliminaries, we classify all the finite groups that can be realized as the full automorphism group of a polarized abelian surface over a finite field in a certain suitable sense.