Date

March 9 (Monday) -- 11 (Wednesday), 2009

Place

3-rd conference room, bldg. 55-S 2F, Okubo campus Waseda University

Supporting Organizations

Institute of Pure and Applied Mathematics of Waseda University
Waseda Institute for Advanced Study (WIAS)

Organizers

YU Jing (National Tsing Hua), KOMATSU Kei-ichi (Waseda), HASHIMOTO Ki-ichiro (Waseda), UMEGAKI Atsuki (WIAS)

10:00--10:50 MORISAWA, Takayuki (Waseda)

A class number problem in the cyclotomic Z₃ extension of Q

Abstract. Let p be a prime number. It is an interesting problem to consider whether a prime number l divides class numbers of the intermediate fields of the cyclotomic Z_p-extension of Q. In the case l =p, Iwasawa proved that p does not divide class numbers of all the n-th layers of the cyclotomic Z_p-extension of Q. Moreover, in the case p=2, Fukuda and Komatsu showed that l does not divide class numbers of all the n-th layers of the cyclotomic Z₂-extension of Q for l <10⁸. In this conference, we talk about the case p=3.

11:10--12:00 AIBA, Akira (Ibaraki)

Analogy of the Weber's class number problem in positive characteristic case (after Gold-Kisilevsky)

13:45 -- 14:45 YAMAMURA, Ken (National Defense Academy)

Showing h(K)=1 or K_ur=K and related topics

Abstract. For some algebraic number fields K, we describe techniques for showing h(K)=1 or K_ur=K and related topics. We present several examples.

15:00--15:50 OKAZAKI, Ryotaro (Doshisha)

On the sizes of relative units of the real abelian number fields of 2-power conductors

Abstract. For n ∈ N, let ζ_n be a primitive 2ⁿ⁺²-th root of unity and K_n=Q(ζ_n+ζ_n^-1). We consider an arbitrary unit ε∈K_n whose relative norm to K_n-1 equals -1. We prove its square Euclidean norm (defined via Minkowski embedding) is larger than or equal to 2ⁿ(2ⁿ⁺¹-1). Then, we will discuss how this inequality is applied in Horie and Fukuda-Komatsu study of Weber's class number problem.

16:00--16:50 ASADA, Mamoru (Kyoto Institute of Technology)

On Galois Groups of abelian extensions of the field generated by l-th roots of unity for all primes l

Abstract. We shall consider the field obtained by adjoining l-th roots of unity for all primes l to a finite algebraic number fied and its abelian extension, the maximal pro-p abelian extension unramified outside p, p being a prime. We shall investigate the structure of this Galois group with the action of the cyclotomic Galois group.

17:00--17:50 KURIHARA, Masato (Keio)

On the Stickelberger ideals for cyclotomic fields

Abstract. This is a joint work with Takashi Miura. I will talk on the exact relation between the Stickelberger ideal defined by Iwasawa and the ideal class group of a cyclotomic field. I will also discuss some related topics on the class groups of CM-fields.

10:00--10:30 KODA, Hideki (Waseda)

On hyperelliptic curves of split type and relations of Jacobstahl sums

Abstract. For a separable polynomial f(x) over the finite field F_q , the sum of the quadratic residues (f(x)/q) for x∈F_q is called a Jacobstahl sum. We shall show, by elementary and direct method, some simple relations between a Jacobstahl sum for certain type of polynomials of degree 8,7,6,5 and those of degree 3.

10:45--11:30 KAWACHI, Mayumi (Tokyo Metropolitan)

Leading coefficients of isogenies of degree p over Q_p

Abstract. Let E be an elliptic curve over Q_p which has potentially supersingular good reduction. Let L/Q_p be a totally ramified extension such that E has good reduction over L and E˜ be the reduction of E mod π where π is a prime element of the ring of integers O_L of L. Let Ê be the formal group over O_L associated to E/O_L. The multiplication by p map [p]: Ê→Ê is written by power series [p](x)=px+c₂x²+…+c_px^p+…+c_p²x^p2+…∈ O^L[[x]]. By using the liftings over O_L of the Dieudonné module of p-divisible group E˜(p) over R_p , we determine the values of v_L(c_p).

13:15--14:00 HASEGAWA, Takehiro (Tsuru)

A tower of function fields related to an arithmetic geometric mean

Abstract. Let K be the finite field of cardinality q² (that is, K=F_q²), and F function fields (of one variable) over K. We denote by g(F) the genus of F and by N(F) the number of degree one places of F/K. A tower of function fields over K is a sequence (F_{0}, F_{1}, F_{2},…) such that the extension F_i+1/F_i is separable for all i and g(F_s)>1 for some s. The result of Drinfeld and Vladut gives that lim_i→∞N(F_i)/g(F_i)≤ q-1. A tower is said to be asymptotically optimal if it attains this bound. Such a tower is used to construct good Goppa codes. In this talk, we define a asymptotically optimal tower related to arithmetic geometric mean, and the ratio lim_i→∞N(F_i)/[F_i:F₀] is equal to the number of the zeroes of a polynomial.

14:10--15:00 NAMIKAWA, Ken-ichi (Osaka)

On mod p non-vanishing of special values of L-functions associated to modular forms over imaginary quadratic fields

Abstract. Let f be a cusp form of GL(2) over rational number field and we take an arbitrary prime p independently of f. There exists a complex number called complex period of f which is determined up to p-adic unit. Then it's known by Shimura that a ratio of the period and the special value of twist of L-function of f by an arbitrary Dirichlet character is an algebraic number. Furthermore, Ash, Stevens and Prasanna proved that there exists a Dirichlet character such that the ratio is actually a p-adic unit. In this talk, we show an analogous result of the latter for cusp forms of GL(2) over imaginary quadratic fields.

15:10--16:00 CHANG, Chieh-Yu (National Tsing Hua)

On periods and logarithms for Drinfeld modules

Abstract. We will discuss some special values occuring from Drinfeld modules, eg. periods, quasi-periods and logarithms of algebraic points. We will also discuss the application of motivic transcendence theory to determine all the algebraic relations among those special values in question.

16:10--17:00 YU, Jing (National Tsing Hua)

On tori and a Galois theory for special Gamma values in positive characteristic

Abstract. We study both geometric gamma values and arithmetic gamma values for rational function fields over finite fields. The motivic Galois group associated to these values are tori over the rational function field in question. We explain how properties of these tori leads to algebraic independence among these gamma values.

10:00--10:50 HASHIMOTO, Ki-ichiro HASHIMOTO (Waseda)

AB-cycles on modular curves

Abstract. It was known that a linear relation of certain ternary theta series attached to orders of a definite quaternion algebra corresponds, throgh Shimura lifting, to a primitive cusp form of weight 2 whose L-functions vanish at s=1. From numerical computations we observed that a linear relation of ternary theta series is always obtained from that of quaternary theta series, which was proved by Arakawa and Böcherer if the level of order is square free. This suggests a possibility (= my dream) of finding 0-cycles on modular curves which give rational points of infinite order on their jacobian varieties.

11:10--12:00 YANG, Yifan (National Chiao Tung)

Construction and application of a class of modular functions

Abstract. In this talk we will present a method to construct modular functions on congruence subgroups of the modular group. We will then discuss several applications of this construction, including defining equations of modular curves, Hauptmoduls of congruence subgroups of genus zero, structure of the cuspidal rational torsion subgroup of the Jacobian J₁(N), and the gonality of the modular curve X₁(N).

13:45--14:35 WANG, Chian-Jen (National Tsing Hua)

A representation theoretic approach to zeta function of complexes

Abstract. Let F be a non-archimedean local field, the zeta function attached to a finite complex arising from Bruhat-Tits building of PGL₃(F) was introduced by Kang and Li. A closed form expression was obtained using combinatorial arguments. In this talk I will discuss an approach using p-adic representation theory to get this closed form, which is a joint work with Kang-Li.

14:55--15:45 KOBAYASHI, Shin-ichi (Nagoya)

Integral structures on p-adic Fourier theory

Abstract. We talk about a generalization of Amice's results on the p-adic distribution on Z_p to the integer ring of arbitrary local field over Q_p based on Schneider-Teitelbaum's p-adic Fourier theory. As applications, we give a measure theoretic proof of Katz's congruence between Bernoulli-Hurwitz numbers at inert primes and a new construction of the Mazur-Tate-Teitelbaum p-adic L-function of CM elliptic curves at supersingular primes.

16:00--16:50 SUZUKI, Masatoshi (Tokyo)

Continuous deformations of the Riemann zeta-function

Abstract. In this talk, two kind of one parameter families of deformations for the Riemann zeta function are introduced referring to the functional equation of the Riemann zeta function. At first, I explain that these two kinds of families are related closely to the distribution zeros of the Riemann zeta-function, and that they are also related to spectral interpretations of the zeros of the Riemann zeta-function. Successively characteristic of each family is discussed, roughly speaking, one of them is arithmetical and another is analytical. After that, a relation of these two families is described.

17:00--17:50 OKANO, Keiji (Waseda)

The commutativity of the maximal unramified pro-p-extensions over the cyclotomic Z_p-extensions

Abstract. Let G be the Galois group of the maximal unramified pro-p-extension of the cyclotomic Z_p-extension over a number fields. The question that what kind of groups can appear as the Galois group G, and what kind of properties characterize G is a mysterious problem. I study the condition for G to be abelian. This talk is a report of the following topics: 1. To give necessary conditions for G to be abelian, 2. To find abelian Galois group G with higher rank, 3. To give examples that a special value of the characteristic polynomial of the Iwasawa module determines whether G is abelian.