The 17th. Joint Workshop on Number Theory between Japan and Taiwan
Date
Saturday, 16th March, 2013 -- Monday, 18th March, 2013
Place
3-rd conference room, bldg. 55-S 2F, Nishiwaseda campus Waseda University
Schedule
| date | 1 | 2 | lunch | 3 | 4 | 5 | Banquet |
|---|---|---|---|---|---|---|---|
| 3/16 (Sat) | 10:00-10:45 | 11:05-11:50 | lunch | 14:00-14:45 | 15:05-15:50 | 16:10-16:55 | �~ |
| 3/17 (Sun) | 10:00-10:45 | 11:05-11:50 | lunch | 14:00-14:45 | 15:05-15:50 | 16:10-16:55 | 18:00-20:00 |
| 3/18 (Mon) | 10:00-10:45 | 11:05-11:50 | lunch | 14:090-14:45 | 15:05-15:50 | �~ | �~ |
Organizers
Keiichi Komatsu (Waseda University),
Kiichiro Hashimoto (Waseda University),
Manabu Ozaki (Waseda University),
Hiroshi Sakata (Waseda University Senior High School)
Supporting Organizations
JSPS Grant-in-Aid for Scienti c Research (C) (24540030,Keiichi Komatsu) and (21540030, Manabu Ozaki).
PDF file
Program�@�@ Abstract
Saturday, 16th March
10:00--10:45 Noriyuki Suwa (Chuo University)
Kummer theory for algebraic tori and normal basis problem
abstract:
We have a rich accumulation in the study on the Kummer theory
for algebraic tori, due to Kida, Hashimoto, Rikuna and others.
In this talk I explain a relation between the Kummer theory
for algebraic tori and the normal basis problem.
We adapt Serre�fs method to formulate the Kummer and
Artin-Schreier-Witt theories by analysis of the group
scheme representing the unit group of a group algebra.
11:05--11:50 Kazuki Sato (Tohoku University)
Hasse principle for the Chow groups of zero-cycles on quadric brations
abstract:
We consider the global-to-local map for the Chow groups of zero-cycles
on varieties over number fields.
For a quadric fibration over a smooth projective curve,
Parimala and Suresh proved that the global-to-local map
(restrected to real places) is injective under a certain assumption if
the dimension is 4 and over.
In this talk, we give a sufficient condition for the injectivity of the
global-to-local map in the case where the
dimension is 2 or 3.
This condition does not imply the injectivity of the global-to-local
map restricted to real places, and we give an example of this.
14:00 -- 14:45 Takahiro Kitajima (Keio University)
On the orders of K-groups of ring of integers in the cyclotomic Zp-extensions of Q
abstract:
J. Coates raised a problem on the boundedness of the class numbers of
intermediate fields of a Zp -extension.
In this talk, we consider the generalization of this problem on
the ideal class groups to that on the torsion part of
K-groups and investigate what happens for the torsion part
of the K-groups of ring of integers.
We especially study the orders of K2m-2 in the cyclotomic
Zp-extension for each positive even
number m and each prime number p.
I will talk on the divisibility of the orders of K2m-2
of ring of integers in the cyclotomic
Zp-extensions of Q.
15:05--15:50 Cornelius Greither (Universitat der Bundeswehr Munchen)
Lower bounds for ranks of class groups coming from Tate sequences
abstract:
Tate sequences are 4-term exact sequences which link an arithmetically
defined Galois module (in this talk, a group of S-units) to an
explicitly defined module, which can be thought of as an
easy approximation of the arithmetic module.
The extension class of a Tate sequence is a very subtle invariant
which comes from class field theory and is hard to grasp.
But fortunately we very often can extract information from
a Tate sequence without knowing the extension class.
This will be shown in two particular situations.
For certain totally real fields K we will find lower bounds
for the rank of the ell-part of Cl(K), and for certain CM fields
we will find lower bounds for the minus part of the ell-part of the
class group These results reprove and partly generalize earlier
results by Cornell and Rosen, and by R. Kucera and the speaker.
The methods are purely algebraic, involving a little cohomology.
16:10--16:55 Hiroki Takahashi (Hiroshima University)
The Iwasawa ��l-invariants in cyclotomic Zp-extensions
abstract:
For a Zp-extension K�� of a
number field K, let Kn be the
intermediate field such that [Kn : K] = pn,
and An the p-part of the
ideal class group of Kn.
Then there exist three invariants ��p,
��p�� Z��0 and ��p �� Z
such that the order of An is
p��pn+��ppn+��p
for all sufficiently large n (Iwasawa�fs class number formula).
For the cyclotomic Zp-extension of cyclotomic fields,
although it was shown that ��p is always zero
by Ferrero-Washington, there are a lot of examples
with ��p > 0 .
On the other hand, for any prime number l �� p, it was shown
that the l-part of the class number in the
cyclotomic Zp-extension is bounded above byWashington.
It was also shown that ��l is bounded above by Friedman.
In a joint work with Ichimura (Ibaraki University) and
Nakajima (Gakushuin University), we calculated ��l(Kn)
for l = 3 and Kn = Q(cos 2��/pn+1, ��3)
in the range p < 600.
I will explain the computation and some results: for all primes p with
5 �� p < 600 and all n �� 0 ,
��3(Kn) = ��3(K0),
0 �� ��3-(Kn) �� 19360,
��3+
(Kn) = 0 and so on.
Sunday, 17th March
10:00--10:45 Yasuo Ohno (Kinki University)
On a property of di-Bernoulli numbers
abstract:
Poly-Bernoulli numbers were defined by
M. Kaneko using poly-logarithms.
He also gave a Clausen-von Staudt type theorem of di-Bernoulli
numbers.
In this talk, we plan to discuss more precise properties of
those numbers modulo small primes.
11:05--11:50 Yoshihiro Onishi (University of Yamanashi)
Explicit realization of Coble's hypersurfaces in terms of multivariated ��-functions
abstract:
Coble�fs hypersurface is known as the cubic hypersurface in a
natural projective space whose singular locus contains the
Jacobian variety of a curve of genus two, or as the biquadratic
hypersurface in a similar space whose locus contains the
Kummer threefold of a curve of genus three.
This is a joint work with J.C. Eilbeck, J. Gibbons,
and E. Previato.
I will present very explicit and universal-type equations
of Coble�fs hypersurfaces by using multivariated ��-
functions.
I will demonstrate the derivatives of our equation with
respect not only to the defining variables but also the
coefficients of equation of the curve give the defining equations
of the Jacobian variety and the Kummer threefold.
As a result, in genus two case, we see that the singular
locus coincides with the Jacobian variety itself.
For any genus three trigonal curve, the singular locus
is very plausible to coincide with the Kummer threefold itself.
14:00--14:45 Takashi Hara (Osaka University)
On Culler-Shalen theory for 3-manifolds and related topics
abstract:
In topology -a research field where one pursuits �gshapes�h
of objects-, it goes without saying that the procedure
�eto decompose manifolds into simpler ones�f
is indispensable.
For 3-manifolds,
in particular, the decomposition along essential surfaces
plays an important role.
Marc Culler and Peter Shalen established in 1983 a method
to construct non-trivial essential surfaces contained
in 3-manifolds in a systematic manner.
There they effectively utilised highly algebraic devices;
for instance, geometry of character varieties
(moduli of 2-dimensional representations of fundamental groups),
theory of trees established by Hyman Bass and Jean-Pierre Serre and
so on.
After brief review on classical Culler-Shalen theory,
we present in this talk an extension of their theory to
higher dimensional character varieties via Bruhat-Tits
theory and (a trial of) an application to arithmetic
topology `a la Barry Mazur et Masanori Morishita.
This is a joint work with Takahiro Kitayama
(the University of Tokyo).
15:05--15:50 Yasushi Mizusawa (Nagoya Institute of Technology)
Iwasawa invariants of links and an analogue of Greenberg's conjecture
(a joint work with Teruhisa Kadokami)
abstract:
Based on the analogy between knots and primes, J. Hillman,
D. Matei and M. Morishita defined the Iwasawa invariants
for cyclic branched covers of links with an analogue of
Iwasawa�fs class number formula.
We consider the existence of covers of links with prescribed Iwasawa
invariants.
We also propose and consider a problem analogous
to Greenberg�fs conjecture.
16:10--16:55 Masanori Morishita (Kyushu University)
Johnson maps in non-Abelian Iwasawa theory
abstract:
We shall introduce arithmetic analogues of Johnson maps
in the context of non-abelian Iwasawa
theory and give their cohomological interpretation.
18:00--20:00 Banquet
Monday, 18th March
10:00--10:45 Noriko Wakabayashi (Kyushu Sangyo University)
Sum formula for mod p multiple zeta values
abstract:
The multiple zeta values, first considered by L. Euler,
are a natural generalization of the values of the Riemann
zeta function at positive integers.
It is known that there are many Q-linear relations among
the values.
The �gsum formula�h is one of the most famous such relations.
The mod p multiple zeta values, with p prime,
have been investigated mainly by M. Hoffman
and refined by D. Zagier.
The main topic here is the �gsum formula�h for
mod p multiple zeta values, which is conjectured
by M. Kaneko and proved by S. Saito and the speaker.
11:05--11:50 Shunsuke Yamana (Kyushu University)
L-functions and theta correspondence for quaternionic unitary groups
abstract:
For any irreducible cuspidal automorphic representation
of quaternionic unitary groups, I will give a necessary
and sufficient condition for its global theta lifting
to be nonvanishing in terms of the analytic properties
of the complete L-function and the occurrence
in the local theta correspondence.
14:00--14:45 Tomoya Machide (Kinki University)
Quadruple zeta values and asymptotic properties of quadruple polylogarithms
abstract:
Asymptotic expansions of multiple polylogarithms are written
in terms of polynomials whose coefficients are multiple zeta values.
In this talk, using asymptotic expansions and identities of
quadruple polylogarithms, we give a parameterized sum
formula of quadruple zeta values.
As applications, we reprove the original sum formula
and some weighted sum formulas.
15:05--15:50 Hiro-aki Narita (Kumamoto University)
Non-vanishing theta lifts to non-split forms of GSp(2)
abstract:
In this talk we provide examples of non-vanishing theta
lifts from an inner form GSO*(4)
of the orthogonal group of degree four to
automorphic forms on the inner forms GSp(1; 1)
and GSp*(2) of the split symplectic
group GSp(2) of degree two, where GSp(1; 1)
(respectively GSp*(2))
denotes the non-split and non-compact
inner form (respectively the compact inner form).
For the case of GSp(1; 1) the method is to find
non-vanishing Bessel periods (or Fourier coefficients)
of the theta lifts.
On the other hand, for the case of
GSp*(2), we can reduce the
problem to non-vanishing of elliptic theta series
attached to some harmonic polynomials.
For the latter we point out that such examples are
given by Ibukiyama-Ihara (Math. Ann. 278).
If time allows, we present an explicit formula for
Bessel periods of the theta lifts to GSp(1; 1) in
terms of the central L-values of some convolution
type L-functions.