Number Theory Workshop at Waseda University
2015 年 2 月 11 日 更新


整数論研究集会ホーム 日本語・概要 (2014) English (2014) お問い合わせ

2014 プログラム

3 月 11 日 (火) 3 月 12 日 (水) 3 月 13 日 (木) 駅から会場まで PDFファイル(プログラム) PDFファイル(アブストラクト)


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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. なお,この研究集会は小松啓一氏(早稲田大学),尾崎学氏 (早稲田大学)の科研費より援助を受けています.


2014 年 3 月 11 日 (火) ~ 13 日 (木)


〒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 N 棟 • 1 階 第 2 会議室


日付 1 2 3 お昼休み 4 5 6 7 Party
3/11 (火) 10:00-10:45 11:05-11:50 × lunch 14:00-14:45 15:05-15:50 16:10-16:5517:15-18:00×
3/12 (水) 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
3/13 (木) 9:30-10:15 10:35-11:20 11:40-12:25 × × × × × ×


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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プログラム   アブストラクト

3 月 11 日 (火)

10:00--10:45 大竹 秀一 (早稲田大学)

Orthogonal decompositions of integral trace forms of cyclotomic fields

Trace form is the symmetric F-bilinear form on K:= F[x]=(f(x)) defined by (x,y) → traceK/F(xy), where f(x) is a separable polynomial over the field F of characteristic different from two.
If f(x) is an irreducible polynomial over the field of rational numbers Q, then the restriction of the trace form to the ring of integers OK of the number field K defines a symmetric bilinear form over the ring of rational integers Z on OK - called the integral trace form of K.
In this talk, we give orthogonal decompositions of integral trace forms of cyclotomic fields and their canonical forms over the ring of p-adic integers explicitly by using Bezoutian forms.

11:05--11:50 藤井 俊 (金沢工業大学)

On Greenberg's conjecture for CM-fields

Let k/Q be a finite extension and p an odd prime number.
Let K/k be the maximal multiple Zp-extension, and let Gal(K/k) [同型] Zpd.
Let L/K be the maximal unramified abelian pro-p extension and X its Galois group.
By a fundamental fact of Iwasawa theory, X is a module over the formal power series ring Λ with coefficients in Zp of d-variables.
Then Greenberg conjectured that X is pseudo-null over Λ.
In this talk, we show that Greenberg’s conjecture holds under the following four conditions:
1) k is a CM-field such that p splits completely.
2) Leopoldt’s conjecture holds for k and p (e.g. k is imaginary abelian).
3) p does not divide the class number of k,
4) λp(k+) = μp(k+) = νp(k+) = 0, where λ, μ and ν denote Iwasawa invariants and k+ denotes the maximal totally real subfield of k.

14:00--14:45 尾崎 学 (早稲田大学)

The Neukirch-Uchida theorem for a certain class of number fields of infinite degree

I will give a Neukirch-Uchida type theorem (that is, the isomorphism class of a field is determined from its absolute Galois group) for a certain class of number fields of infinite degree.

15:05--15:50 星 裕一郎 (京都大学)

Reconstruction of a Number Field From the Absolute Galois Group

It follows from the Neukirch-Uchida Theorem that the isomorphism class of a number field is completely determined by the isomorphism class of the associated absolute Galois group.
On the other hand, the Neukirch-Uchida Theorem (as well as its proof) does not give a ”functorial grouptheoretic algorithm” for reconstructing the original number field from the absolute Galois group.
In this talk, I discuss such a ”functorial group-theoretic algorithm”.

16:10--16:55 佐久川 憲児 (大阪大学)

A control theorem for the torsion Selmer pointed set

Selmer groups are important arithmetic invariants of Galois representations.
Minhyong Kim defined the Selmer variety which is a
non-abelian analogue of the Qp-Selmer group.
In this talk, we give a torsion analogue of the Selmer variety.
Then, we establish an analogue of Mazur’s control theorem for this torsion analogue.

17:15--18:00 中村 健太郎 (北海道大学)

Local ε-isomorphisms for rank two p-adic representations of Gal((Q_p)^ー/Q_p) and a functional equation of Kato's Euler system

Local ε-isomorphisms are conjectural bases of the determinants of the Galois cohomologies of p-adic representations of Gal(Qp/Qp) which p-adically interpolate local constants (ε-constatnts, L-constants, etc.) associated to de Rham representations.
Up to now, such bases have been constructed for rank one case by Kazuya Kato, crystalline case by Benois-Berger, Loeffler-Zerbes-Venjakob, and trianguline case by the speaker.
In this talk, using Colmez’s theory of p-adic Langlands correspondence for GL2(Qp), we define such bases for (almost) all rank two torsion p-adic representations.
We show that our ε-isomorphisms satisfy the desired interpolation property in many important cases.
As an application, we prove a functional equation of Kato’s Euler systems associated to modular forms without any condition at p.
Under the assumption that Kato’s Euler systems give the zeta isomorphisms, this functional equation implies Kato’s global ε-conjecture.

3 月 12 日 (水)

10:00--10:45 都筑 正男 (上智大学)

Equidistribution and subconvexity bound related to certain L-values

This is joint work with Singo Sugiyama (Osaka Univ.).
I would like to report our recent refinement on a spectral equidistribution result in the level aspect for Satake parameters of holomorphic Hilbert cusp forms weighted by central L-values, and a bound of quadratic base change L-functions for Hilbert cusp forms with a subconvex exponent in the weight aspect.

11:05--11:50 青木 宏樹 (東京理科大学)

Determination of the structure of vector valued Siegel modular forms by using Jacobi forms

In general, the determination of the structure of modular forms is difficult, although the dimension formula is well known.
However, sometimes by using Jacobi forms or Witt operators, we can easily determine the structure of some kinds of modular forms.
In this talk, I shall introduce this strategy and show some examples, including the structure theorem of vector valued Siegel modular forms of level 2.

14:00--14:45 Thomas Wieber(Heidelberg 大学)

Geometrically proven structure theorems for vector valued Siegel modular forms

I shall begin with classical results on vector valued (cuspidal) Siegel modular forms.
Afterwards, I shall present new structure theorems for vector valued Siegel modular forms with respect to Sym2 and Igusa’s subgroup Γ2[2, 4]. They rest on the well known fact that Γ-invariant tensor fields on the Siegel upper halfplane can be viewed as vector valued Siegel modular forms with respect to this group Γ.
For our group the Satake compactification is the 3-dimensional projective space.
After observing the tensors on the Satake compactification the structure theorem(s) and Hilbert function(s) for the representation Sym2 become rather evident.
Here, we discovered a new strategy to retrieve structure theorems for other appropriate groups.
Examples executed by Freitag, Salvati Manni and partially the speaker include the groups of genus two Γ2[4, 8] and Γ2[2, 4, 8] and even one of Igusa’s subgroups of genus 3 Γ3[2, 4].
Using invariant theory we could reprove Aoki’s structure theorem for Γ2,0[2] and Clery’s, van der Geer’s and Grushevsky’s structure theorem for Γ2[2] and Sym2.

15:05--15:50 伊吹山 知義 (大阪大学)

Construction of liftings to vector valued Siegel modular forms

Using the Hayashida-Maass relation of Ikeda lift and good differential operators, we construct several liftings to vector valued Siegel modular forms of integral or half-integral weight from a pair of elliptic modular forms.

16:10--16:55 足立 恒雄 (早稲田大学)

'sacred' or 'profane' ?

What is mathematics? We present the following tentative answer for discussion.
Mathematics takes a syntactical and a semantical form; the syntactical form is usually called pure mathematics, the semantical form applied mathematics.

18:00--20:00 懇親会

3 月 13 日 (木)

9:30--10:15 坂田 実加 (近畿大学)

多重ベルヌーイ数の p-order について

Poly-Bernoulli numbers were introduced and studied by M.Kaneko as a generalization of classical Bernoulli numbers.
He clarified the p-divisibility of denominators of di-Bernoulli numbers.
On the other hand, poly-Bernoulli numbers of negative index have combinatorial interpretation.
In this talk, we plan to discuss their p-divisibility and periodicity.

10:35--11:20 広瀬 稔 (京都大学)

On the theory of fans and its application to Shintani L-function and Hecke L-function

Shintani L-function is a holomorphic function of several variables defined by a certain Dirichlet series.
A fan is a formal sum of cone regions.
I explain about the theory of fans and its application to Shintani L-function and Hecke L-function.

11:40--12:25 兵藤 史武 (早稲田大学)

A formal power series of a Hecke ring associated with the Heisen- berg Lie algebra

This talk studies a formal power series with coefficients in a Hecke ring associated with the Heisenberg Lie algebra.
We relate the series to the classical Hecke series defined by E. Hecke, and prove that the series has a property similar to the rationality theorem of the classical Hecke series.
And then, our results recover the rationality theorem of the classical Hecke series.