The 17th. Joint Workshop on Number Theory between Japan and Taiwan
Saturday, 16th March, 2013 -- Monday, 18th March, 2013
Keiichi Komatsu (Waseda University),
Kiichiro Hashimoto (Waseda University),
Manabu Ozaki (Waseda University), Hiroshi Sakata (Waseda University Senior High School)
JSPS Grant-in-Aid for Scientic Research (C) (24540030,Keiichi Komatsu) and (21540030, Manabu Ozaki).
Saturday, 16th March
10:00--10:45 Noriyuki Suwa (Chuo University)
Kummer theory for algebraic tori and normal basis problem
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 Serrefs 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
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
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
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 Él-invariants in cyclotomic Zp-extensions
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 Ép, Êp¸ Z0 and Ëp ¸ Z such that the order of An is pÉpn+Êppn+Ëp for all sufficiently large n (Iwasawafs class number formula).
For the cyclotomic Zp-extension of cyclotomic fields, although it was shown that Êp is always zero by Ferrero-Washington, there are a lot of examples with Ép > 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 Él is bounded above by Friedman.
In a joint work with Ichimura (Ibaraki University) and Nakajima (Gakushuin University), we calculated Él(Kn) for l = 3 and Kn = Q(cos 2Î/pn+1, Ä3) 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 , É3(Kn) = É3(K0), 0 É3-(Kn) 19360, É3+ (Kn) = 0 and so on.
Sunday, 17th March
10:00--10:45 Yasuo Ohno (Kinki University)
On a property of di-Bernoulli numbers
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
Coblefs 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 Coblefs 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
In topology -a research field where one pursuits gshapesh of objects-, it goes without saying that the procedure eto decompose manifolds into simpler onesf 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)
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 Iwasawafs class number formula.
We consider the existence of covers of links with prescribed Iwasawa invariants.
We also propose and consider a problem analogous to Greenbergfs conjecture.
16:10--16:55 Masanori Morishita (Kyushu University)
Johnson maps in non-Abelian Iwasawa theory
We shall introduce arithmetic analogues of Johnson maps in the context of non-abelian Iwasawa theory and give their cohomological interpretation.
Monday, 18th March
10:00--10:45 Noriko Wakabayashi (Kyushu Sangyo University)
Sum formula for mod p multiple zeta values
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 formulah 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 formulah 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
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
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)
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.