## The 14th. Joint Workshop on Number Theory between Japan and Taiwan

### 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)

## March 9 (Mon)

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

A class number problem in the cyclotomic **Z**_{3} 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 **Z**_{2}-extension of **Q** for *l* <10^{8}.
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 2^{n+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 2^{n}(2^{n+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.

## March 10

### 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+c_{2}x^{2}+…+c_{p}x^{p}+…+c_{p2}x^{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 q^{2} (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}:F_{0}] 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.

## March 11

### 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 J_{1}(N), and the gonality of the modular curve X_{1}(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 PGL_{3}(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.