日程

2010 年 3 月 17 日 (水) ～ 19 日 (金)

会場

〒169-8555　東京都新宿区大久保3-4-1
早稲田大学西早稲田キャンパス（旧・大久保キャンパス） • 55 号館 S 棟 • 2 階 第 3 会議室
地下鉄副都心線「西早稲田駅」からのアクセス

講演時間

日付	1	2	お昼休み	3	4	5	6
3/17 (Wed)	10:00-10:45	11:00-11:45	lunch	13:30-14:15	14:30-15:00	15:15-16:00	16:15-17:00
3/18 (Thu)	10:00-10:45	11:00-11:30	lunch	13:15-14:00	14:15-15:00	15:15-16:00	16:15-17:00
3/19 (Fri)	10:00-10:30	10:45-11:30	lunch	13:30-14:00	14:15-15:00	15:15-16:00	16:15-17:00

研究代表者

Jing YU (国立台湾大学), 小松 啓一 (早稲田大学), 橋本 喜一朗 (早稲田大学), 梅垣 敦紀 (WIAS)

協力

早稲田大学高等研究所 (WIAS)

10:00--10:45 足立恒雄 (早稲田大学)

代数学と選択公理

アブストラクト：ZFとBGという二つの集合論におけるクラス（類） という概念の使い方を紹介する。

11:00--11:45 岡崎龍太郎 (同志社大学)

Diophantine Method for Weber's Class Number Problem

Abstract. Let K_n=Q(cos(2π/2^(n+2))). Denote by h_n its ideal class number. Weber showed h₁ = h₂ =h₃=1 and asked whether h_n=1 holds for every n≧1. Later, Bauer and Masley showed h₄=1 and Linden showed h₅=1. Recently K.Horie initiated a project of proving this conjecture of Weber's. Fukuda and Komatsu made a further progress. Their method is based on the identity between the ideal class number and the index of cyclotomic units. A general idea is that a lower bound for units leads to an upper bound for the class number. The mentioned authors investigated the structure of units in more detail so that they obained a method for handling each possible prime divisor. Lower bounds on units remain essential in the project. In this talk, we will give lower bounds for units in K_n whose norm to K_n-1 equals ±1. Then, we will discuss their application in Weber's class number problem. Amazingly, the arguments are more Diophantine than algebraic or analytic. In the same conference, Morisawa will talk about generalization to other cyclotomic fields.

13:30--14:15 堀江邦明 (東海大学)・堀江充子(お茶の水女子大学)

いくつかの素数pに対する Q 上のZ_p-拡大体の狭義イデアル類群について

14:30--15:00 森澤貴之 (早稲田大学)

Mahler測度とWeberの類数問題

Abstract. (This is joint work with R. Okazaki.) Let p be a prime number. It is an interesting problem to consider whether the class number is equal to one for all intermediate fields of the cyclotomic Z_p-extension of Q. This problem is called "Weber's Class Number Problem". However, direct calculation of the class number is very difficult. So we study the problem to ask whether a prime number l divides those class numbers. In the case p=2, R. Okazaki developed a theory for this problem by using Mahler measure. In this conference, we will present the result of the case where p is an odd prime number.

15:15--16:00 河本史紀（学習院大学）・冨田耕史（名城大学）

Continued fractions and Gauss' class number problem for real quadratic fields

16:15--17:00 水澤靖(名古屋工業大学)・尾崎学(近畿大学)

基本Z_p拡大上の馴分岐pro-pガロア群について

10:00--10:45 佐々木義卓 (近畿大学)

多重高次Mahler測度とWittenの体積公式について

アブストラクト： Mahler測度は、複素数係数のLaurent多項式に 対して定まる量で、ある多項式に対するMahler測度は、 ゼータ関数やL関数の特殊値として記述されることが知られている。 近年、黒川・Lalin・落合によって、Mahler測度を拡張した 多重高次Mahler測度が導入され、その具体例の一つとして、 多重高次Mahler測度と多重ゼータ値との関係が報告された。 本講演では、多重高次Mahler測度とWittenゼータ値と呼ばれる ある多重ゼータ関数の特殊値との関係を述べ、さらに、 ある多項式族の多重高次Mahler測度は、あるモジュライ空間の 体積（Wittenの体積公式）として解釈されることを述べる。

11:00--11:30 Fang-Ting TU (National Chiao Tung Univ.)

Finite graphs and orders of M₂(k) over local field k

13:15--14:00 繁木伸孝（九州大学），金子昌信（九州大学）

楕円モジュラー j 関数の実二次点での「値」とマルコフ 二次無理数

14:15--15:00 Ming-Lun HSIEH [謝銘倫] (National Taiwan Univ)

On the non-vanishing of Hecke L-values modulo p

Abstract. By Zariski density of CM poins in Hilbert modular varieties, Hida established an analogue of Washington's theorem on Hecke L-values for CM fields of split conductor. In this talk we present a similar result for Hecke characters attached to CM abelian varieties over totally real fields based on Hida's ideas and discuss the application to Iwasawa main conjecture for CM fields.

15:15--16:00 Jing YU [于靖] (National Taiwan Univ)

Definite quaternion algebras over function fields, and Brandt matrices

Abstract. Let k be the rational function field over a fixed finite field. We consider quaternion algebra Q over k which ramifies at the infinite place and another finite place P. Given maximal order in Q, the left ideal classes parametrizes rank 2 supersingular Drinfeld modules of characteristic P. One introduces Brandt matrices B(m) parametrized by monic polynomials m in k. In joint work with Fu-Tsun Wei, we derive analogues of Eichler's trace formula, which is expressed in terms of Hurwitz class numbers of imaginary quadratic function fields over k. These traces is further identified with traces of Hecke operators on automorphic forms of Drinfeld type over the function field k.

16:15--17:00 Nikolajs VOLKOVS (University of Toronto)

A new digital signature algorithm with a generator kept secret

10:00--10:30 兵藤 史武（早稲田大学）

ある種の羃ゼロ群のゼータ関数の明示形

アブストラクト： 一般の群に対してゼータ関数を考えることができる. ランク有限の自由アーベル群やHeizenberg群のときは リーマンゼータ関数の積でかけることが知られている. これを一般化することを考えたい.

10:45--11:30 津嶋貴弘（東京大学）

モジュラー曲線 X₀(p^4) の安定モデル

13:30--14:00 YaoHan CHEN (National Chiao Tung Univ.)

Cuspidal Q-rational Torsion subgroup of J(Γ) of Level p

14:15--15:00 加川 貴章（立命館大学）

The Diophantine equation X^3=u+v over real quadratic fields

Abstract. Let k be a real quadratic field. The Diophantine equation X^3=u+v in X∈ O_k (the ring of integers of k), v,v∈ O_k^×(the group of units of k) is investigated under certain assumption of k.

15:15--16:00 市原 由美子 (広島大学）

保型L関数の critical line 上の点における non-vanishing について

Abstract. In 1995, Duke considered the number of cusp forms whose L-functions do not vanish at the central point. He obtained the lower bound of it for the orthogonal cusp forms of weight 2 and level p (p is a prime number). By Duke's method, Akbary studied such type of lower bound for the orthogonal new forms of level p, Kamiya studied such type of lower bound for the orthogonal basis of a space of cusp forms in general. In this talk, I explain such type of lower bound for the orthogonal new forms whose weight is k satisfying 0 < k < 12 or k=14 and level is a power of p.

16:15--17:00 大井 周 (早稲田大学)

2変数形式的KZ方程式と多重ゼータ値の調和積

Abstract. We talk about an algebra structure and a connection problem of the formal KZ equation of two variables via iterated integrals. The fundamental solution to the equation decomposes to a product of two factors which are solutions to the formal KZ equation of one variable. Comparing the different ways of decomposition gives functional relations of the hyperlogarithms which contain the harmonic product of multiple zeta values as a limit. This is a joint work with Kimio Ueno.