## The 15th. Joint Workshop on Number Theory between Japan and Taiwan

### Date

March 17 (Wednesday) -- 19 (Friday), 2010

### Place

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

### Schedule

date | 1 | 2 | lunch | 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 |

### Organizers

YU Jing (National Taiwan Univ.), KOMATSU Kei-ichi (Waseda Univ.), HASHIMOTO Ki-ichiro (Waseda Univ.), UMEGAKI Atsuki (WIAS)

### Supporting Organizations

Waseda Institute for Advanced Study (WIAS)

## March 17 (Wednesday)

### 10:00--10:45 ADACHI Norio (Waseda Univ.)

Algebra and the Axiom of Choice

Abstract. We give two proofs of Steiniz' Theorem; one resorting to the idea of classes in ZF, and the other to the global Axiom of Choice in BG.

### 11:00--11:45 OKAZAKI Ryotaro (Doshisha Univ.)

Diophantine Method for Weber's Class Number Problem

Abstract.
Let **K**_{n}=**Q**(cos(2/2^(n+2))).
Denote by *h*_{n} its ideal class number.
Weber showed *h*_{1} = *h*_{2}
=*h*_{3}=1 and
asked whether *h*_{n}=1 holds for every *n*1.
Later, Bauer and Masley showed *h*_{4}=1 and Linden showed
*h*_{5}=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 HORIE Kuniaki (Tokai Univ.), HORIE Mitsuko (Ochanomizu Univ.)

On the narrow class groups of the **Z**_{p}-extensions over
**Q** for several primes *p*

### 14:30--15:00 MORISAWA Takayuki (Waseda Univ.)

Mahler Measure and Weber's Class Number Problem

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 KAWAMOTO Fuminori (Gakushuin Univ.), TOMITA Koshi (Meijo Univ.)

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

### 16:15--17:00 MIZUSAWA Yasushi (Nagoyakougyo Univ.), OZAKI Manabu (Kinki Univ.)

On tame pro-p Galois groups over basic **Z**_{p}-extensions

## March 18 (Thursday)

### 10:00--10:45 SASAKI Yoshitaka (Kinki Univ.)

On multiple Mahler measure of higher degree and Witten's volume formula

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

Finite graphs and orders of *M _{2}(k)* over local field

*k*

### 13:15--14:00 SHIGEKI Nobutaka (Kyushu Univ.), KANEKO Masanobu (Kyushu Univ.)

Values of elliptic *j*-functions at real quadratic points and Markov quadratic irrational numbers

### 14:15--15:00 HSIEH Ming-Lun (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 YU Jing (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 VOLKOVS Nikolajs (Univ. of Toronto)

A new digital signature algorithm with a generator kept secret

## March 19 (Friday)

### 10:00--10:30 HYODO Fumitake (Waseda Univ.)

Explicit form of zeta-functions of some nilpotent groups

Abstract: For general groups, we can consider zeta functions as the generating function of the number of subgroups of given index n. If G is a free abelian group of finite rank or the Heizenberg group, it is known that the zeta function associated to G forms the multiplication of shifts of the Riemann zeta functions. We will present the generalizetion of this result.

### 10:45--11:30 TSUSHIMA Takahiro (Univ. of Tokyo)

Stable model of the modular curves *X _{0}(p^4)*

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

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

### 14:15--15:00 KAGAWA Takaaki (Ritsumeikan Univ.)

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 ICHIHARA Yumiko (Hiroshima Univ.)

Non-vanishing of the automorphic *L*-functions at the point on the critical line

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 OI Shu (Waseda Univ.)

The formal *KZ* equation of two variables and harmonic product
of multiple zeta values

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.