ASARC 정수론 워크샵

일시 :  2013/12/16-18

장소 : 해운대 리베라 호텔

Time Table

	16(Mon)	17(Tue)	18(Wed)
10:00-10:50		A. Schweizer	최 소 영
11:00-11:30		최 윤 서	진 석 호
11:40-12:10		장 승 환	이 동 욱
12:10-14:00	Check in	점 심
14:00-14:30	구 자 경	이 준 호
14:40-15:10	이 상 준	정 환 엽
15:30-16:00	최 도 훈	오 동 열
16:10-17:00	이 정 연	차 병 철

초록

구자경 (KAIST)

Title: Ring class invariants by smaller generators

Abstract: Over imaginary quadratic fields we investigate ring class invariants in terms of eta-quotient whose minimal polynomials have relatively small coefficients.

이상준 (KAIST)

Title:  Universality of random graphs for graphs of maximum degree two.

Abstract: A graph $G=G(V,E)$ consists of a vertex set $V$ and an edge set $E$ of two element sets of $V$. A complete graph is a graph in which each pair of two vertices are linked by an edge. For a positive integer $n$ and a real number $p$ in the range $0\leq p\leq 1$, the random graph $G(n,p)$ on a set $V$ of $n$ elements can be obtained from the complete graph on $V$ by choosing each edge with probability $p$, independently of all other edges. After the random graph $G(n,p)$ was first introduced by Erd\H{o}s in 1947, the theory of the random graph has become an active area of research. One of the most interesting problems is the containment problem, in which one tries to obtain conditions on $p$ for the property that $G(n,p)$ contains a given graph $H$ as a subgraph with high probability.

In this talk, we consider the following containment problem: For a family $\cF$ of graphs, a graph $G$ is called \emph{$\cF$-universal} if $G$ contains every graph in $\cF$ as a subgraph. Let $\cF^*$ be the family of all graphs on $n$ vertices with maximum degree at most $2$. We show that the random graph $G(n,p)$ is  $\cF^*$-universal with high probability provided

$p\geq C\big(\frac{\log n}{n}\big)^{1/2}$ for a sufficiently large constant~$C$.

This is joint work with Jeong Han Kim (KIAS).

최도훈 (한국항공대)

Title: Congruences for Fourier coefficients of a modular form

Abstract: The Fourier coefficients of a modular form are related to other objects in number theory such as the number of points of an elliptic over a finite field and the values of the partition function and so on. In this talk, I will talk about arithmetic property of Fourier coefficients of a modular form especially in terms of their congruences.

이정연 (KIAS)

Title: Integrality of zeta values and its arithmetic applications

Abstract:

In this talk, we introduce two kinds of integrality of partial zeta

values and discuss their applications to arithmetic. The first integrality indicates that the denominator of partial zeta values at s = -k has a bound determined only by k. It means that the denominator does not varies where associated elds of zeta functions varies. I am going to present how this integrality is efficiently used in class number problems of real quadratic fields. The second is the one for l-smoothed partial zeta function: the values of the function at s = -k is integer. This is a crucial ingredient in building p-adic l-functions by interpolating the values at nonpositive integers. This appears in relation to annihilator of class group of totally real number fields, or more generally its higher analogue. From the congruence between values at s = -k and those at s = 0 of l-smoothed zeta function which is called Coates-Sinnot congruence, one relates annihilator of class group and annihilator of higher K-group of totally real fields.

A. Schweizer (KAIST)

Title:  Elliptic points of the Drinfeld modular groups

Abstract: As a function field analog of the action of the classical modular group $SL_2({\mathbb Z})$ on the complex upper halfplane, the Drinfeld modular group $GL_2(A)$ acts on the Drinfeld upper halfplane. Here $A$ is a polynomial ring ${\mathbb F}_q[t]$ or a more complicated ring of integers in a function field. We describe the elliptic points of this action of $GL_2(A)$, that is, the points whose stabilizer is bigger than the center of  $GL_2(A)$. This is related to a certain class group. It also has implications for the action of $GL_2(A)$ on the Bruhat-Tits tree, which say something about the group-theoretic structure of $GL_2(A)$. This is joint work with Alec Mason from Glasgow.

최윤서 (KIAS)

Title: theta functions, Riemann theta functions, and elliptic functions

Abstract: The definitions and the properties of theta functions and elliptic functions are introduced. Also, some results related to the theta functions in Ramanujan's work are briefly introduced. Then, we discuss about the cubic analogues of theta functions and elliptic functions

장승환 (이화여대)

Title: Elliptic curves over finite fields and pairings

Abstract: I will talk about certain problems in cryptography that are related to

elliptic curves defined over finite fields and bilinear pairings on them.

이준호 (KIAS)

Title: On totally real cubic orders generated by a unit

Abstract: We introduce known results on fundamental units of quadratic, cubic, and quartic orders. In particular, we make a close investigation on fundamental units of cubic orders. This is a joint work with Stephane. Louboutin.

정환엽 (충북대)

Title:  Average values of L-series for the prime hyperelliptic ensemble

Abstract: In this talk, we will talk on asymptotic formulae for the first and second moments of quadratic Dirichlet L–functions of prime conductor

오동렬 (포항공대)

Title: Valuation overrings of a Noetherian domain

Abstract: Let R be a Noetherian domain and 0 = [eqn: P _{0}] ⊊ [eqn: P _{1}] ⊊ [eqn: CDOTS] ⊊ [eqn: P_n] be a saturated chain of prime ideals of R. Let V be a valuation overring of R that has a chain of prime ideals [eqn: \{Q _alpha \} _{a IN LAMBDA }}] such that [eqn: \{rm Q _{alpha } SMALLINTER R rm\} _{alpha IN LAMBDA }]= {[eqn: {P _{i}\}_{i =0}^{n}] In this talk, we prove that [eqn: \{Q _alpha \} _{a IN LAMBDA }}]={[eqn: 0=Q _{0}\ ⊊ \ Q _{1} \ ⊊ \ CDOTS ⊊ \ Q_n]}  and [eqn: V_Q_n]is discrete, i.e., [eqn: Q_i V_Q_i] is principal for all [eqn: i=1, LDOTS , n]. We say that R satises property (*) if [eqn: R _{P}] is a Noetherian domain for each prime ideal P of R with htP < [eqn: INF]. We also show that if {[eqn: P_n]} is a chain of prime ideals of an integral domain R with property (*) such that htP_n < [eqn: INF] for each n, then there exists a discrete valuation overring of R which has a chain of prime ideals lying over {[eqn: P_n]}.

차병철 (Muhlenberg College)

Title: Prime number races on elliptic curves in function fields

Abstract: Fix an elliptic curve $E$ over rationals without CM. For each prime $p$ of good reduction, let $a_p$ be the usual trace of Frobenius at $p$. "Prime number race on elliptic curves" is concerned with the limiting distribution of the function \[ S(x) := \#\{ p \le x \mid a_p < 0 \} - \#\{ p \le x \mid a_p > 0\} \] as $x\to\infty$. We investigate this question and its variation in the

context of function fields. This is joint work with Daniel Fiorilli and Florent Jouve.

최소영(동국대)

Title: Valence formulas for certain arithmetic groups and their applications

Abstract: We find valences formulas related with arithmetic groups $\Gamma^{+}_{0}(N)$. As an application, we find an upper bound of the dimension of the space of modular forms for $\Gamma^{+}_{0}(N)$. Moreover Sturm bounds and certain linear relations for cuspidal Poincar\'e series are derived. This is a jonit work with Chang Heon Kim.

진석호 (포항공대)

Title: Periods of Jacobi forms and Hecke operator

Abstract: A Hecke action on the space of periods of cusp forms, that is compatible with that on the space of cusp forms, was first computed by Y. Manin and an explicit algebraic formula of Hecke operators acting on the space of period functions of modular forms was derived by studying the rational period functions by Choie-Zagier. As an application an elementary proof of the Eichler–Selberg trace formula was derived by Zagier. A analogous modification has been applied to the space of period functions of Maass cusp forms also.

We study the space of period functions of Jacobi forms by means

of the Jacobi integral and give an explicit description of the action of Hecke operators on this space. A Jacobi–Eisenstein series of weight 2 and index 1 is discussed as an example. Periods of Jacobi integrals already appeared in a disguised form in the work of Zwegers in his study of the Mordell integral coming from Lerch sums, and mock Jacobi forms are typical examples of the Jacobi integral.

이동욱 (포항공대)

Title: Non-emptiness of Newton stratification of Shimura varieties and the Langlands-Rapoport conjecture.

Abstract: The Newton stratification on a special fiber of a Shimura variety (of abelian type) is a stratification defined in terms of the slope sequence of the Frobenius action on the Dieudonne module (``F-isocrystals with G-structure"). There has been a conjectural group-theoretic description of the F-isocrystals that are expected to show up in a given Shimura variety.

We show how this conjecture follows from the Langlands-Rapoport conjecture. The latter conjecture, which was recently confirmed by M. Kisin, provides a group-theoretic description of the set of the points over finite fields of a (good) reduction of Shimura variety.

This talk will be also a brief introduction to the Langlands-Rapoport conjecture, which is an important big step in the Langlands program of expressing the Hasse-Weil L-function of a Shimura variety in terms of automorphic L-functions.

Memo