## 2013早稲田整数論研究集会

早稲田大学数理科学研究所の研究事業の一環として，下記のように研究集会を催 しますので，ご案内申し上げます．

### 日程

2013 年 3 月 16 日 （土） ～ 18 日 （月）

### 会場

〒169-8555 東京都新宿区大久保3-4-1

早稲田大学西早稲田キャンパス（旧・大久保キャンパス） •
55 号館 S 棟 • 2 階 第 3 会議室

地下鉄副都心線「西早稲田駅」からのアクセス

### 講演時間

日付 | 1 | 2 | お昼休み | 3 | 4 | 5 | Party |
---|---|---|---|---|---|---|---|

3/16 （土） | 10:00-10:45 | 11:05-11:50 | lunch | 14:00-14:45 | 15:05-15:50 | 16:10-16:55 | × |

3/17 （日） | 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 （月） | 10:00-10:45 | 11:05-11:50 | lunch | 14:00-14:45 | 15:05-15:50 | × | × |

### 研究代表者

小松 啓一 （早稲田大学），橋本 喜一朗 （早稲田大学），
尾崎 学 （早稲田大学），

坂田 裕 （早稲田大学高等学院）

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## 3 月 16 日 （土）

### 10:00--10:45 諏訪 紀幸 （中央大学）

Kummer theory for algebraic tori and normal basis problem

アブストラクト：

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’s 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 佐藤 一樹 （東北大学）

Hasse principle for the Chow groups of zero-cycles on quadric brations

アブストラクト：

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 北島 孝浩 （慶應義塾大学）

On the orders of *K*-groups of ring of integers
in the cyclotomic *Z _{p}*-extensions of

*Q*

アブストラクト：

J. Coates raised a problem on the boundedness of the class numbers of
intermediate fields of a *Z _{p}* -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

*K*in the cyclotomic

_{2m-2}*Z*-extension for each positive even number

_{p}*m*and each prime number

*p*.

I will talk on the divisibility of the orders of

*K*of ring of integers in the cyclotomic

_{2m-2}*Z*-extensions of

_{p}*Q*.

### 15:05--15:50 Cornelius Greither (Universitat der Bundeswehr Munchen)

Lower bounds for ranks of class groups coming from Tate sequences

アブストラクト：

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 高橋 浩樹 （広島大学）

The Iwasawa *λ _{l}*-invariants in cyclotomic

*Z*-extensions

_{p}
アブストラクト：

For a *Z _{p}*-extension

*K*of a number field

_{∞}*K*, let

*K*be the intermediate field such that

_{n}*[K*, and

_{n}: K] = p^{n}*A*the

_{n}*p*-part of the ideal class group of

*K*.

_{n}Then there exist three invariants

*λ*,

_{p}*μ*and

_{p}∈ Z_{≧0}*ν*such that the order of

_{p}∈ Z*A*is

_{n}*p*for all sufficiently large

^{λpn+μppn+νp}*n*(Iwasawa’s class number formula).

For the cyclotomic

*Z*-extension of cyclotomic fields, although it was shown that

_{p}*μ*is always zero by Ferrero-Washington, there are a lot of examples with

_{p}*λ*.

_{p}> 0On the other hand, for any prime number

*l ≠ p*, it was shown that the

*l*-part of the class number in the cyclotomic

*Z*-extension is bounded above byWashington.

_{p}It was also shown that

*λ*is bounded above by Friedman.

_{l}In a joint work with Ichimura (Ibaraki University) and Nakajima (Gakushuin University), we calculated

*λ*for

_{l}(K_{n})*l = 3*and

*K*in the range

_{n}= Q(cos 2π/p^{n+1}, ζ_{3})*p < 600*.

I will explain the computation and some results: for all primes

*p*with

*5 ≦ p < 600*and all

*n ≧ 0 , λ*and so on.

_{3}(K_{n}) = λ_{3}(K_{0}), 0 ≦ λ_{3}^{-}(K_{n}) ≦ 19360, λ_{3}^{+}(K_{n}) = 0## 3 月 17 日 （日）

### 10:00--10:45 大野 泰生 （近畿大学）

On a property of di-Bernoulli numbers

アブストラクト：

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 大西 良博 （山梨大学）

Explicit realization of Coble's hypersurfaces in terms of multivariated п-functions

アブストラクト：

Coble’s 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’s hypersurfaces by using multivariated п-
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 原 隆 （大阪大学）

On Culler-Shalen theory for 3-manifolds and related topics

アブストラクト：

In topology —a research field where one pursuits “shapes”
of objects—, it goes without saying that the procedure
‘to decompose manifolds into simpler ones’
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 水澤 靖 （名古屋工業大学）

Iwasawa invariants of links and an analogue of Greenberg's conjecture

（門上晃久氏との共同研究）

アブストラクト：

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’s 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’s conjecture.

### 16:10--16:55 森下 昌紀 （九州大学）

Johnson maps in non-Abelian Iwasawa theory

アブストラクト：

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 懇親会

## 3 月 18 日 （月）

### 10:00--10:45 若林 徳子 （九州産業大学）

Sum formula for mod *p* multiple zeta values

アブストラクト：

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 “sum formula” 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 “sum formula” for
mod *p* multiple zeta values, which is conjectured
by M. Kaneko and proved by S. Saito and the speaker.

### 11:05--11:50 山名 俊介 （九州大学）

*L*-functions and theta correspondence for quaternionic unitary groups

アブストラクト：

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 町出 智也 （近畿大学）

Quadruple zeta values and asymptotic properties of quadruple polylogarithms

アブストラクト：

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 成田 宏秋 （熊本大学）

Non-vanishing theta lifts to non-split forms of *GSp(2)*

アブストラクト：

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*of the split symplectic group

^{*}(2)*GSp(2)*of degree two, where

*GSp(1; 1)*(respectively

*GSp*) denotes the non-split and non-compact inner form (respectively the compact inner form).

^{*}(2)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*, we can reduce the problem to non-vanishing of elliptic theta series attached to some harmonic polynomials.

^{*}(2)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.