## 代数学コロキウム

開催情報 水曜日　17:00～18:00　数理科学研究科棟(駒場) 056号室 今井 直毅, 三枝 洋一

### 2011年05月18日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室

On the linear independence of values of some Dirichlet series (JAPANESE)
[ 講演概要 ]

2000年にT.Rivoalにより証明されており,今回の結果はその一般化に相当する.

### 2011年05月11日(水)

17:30-18:30   数理科学研究科棟(駒場) 056号室
Michel Raynaud 氏 (Universite Paris-Sud)
Permanence following Temkin (ENGLISH)
[ 講演概要 ]
When one proceeds to a specialization, the good properties of algebraic equations may be destroyed. Starting with a bad specialization, one can try to improve it by performing modifications under control. If, at the end of the process, the initial good properties are preserved, one speaks of permanence. I shall give old and new examples of permanence. The new one concerns the relative semi-stable reduction of curves recently proved by Temkin.

(本講演は「東京パリ数論幾何セミナー」として、インターネットによる東大数理とIHESとの双方向同時中継で行います。)

### 2011年04月27日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室

Sturm の定理の Hilbert 保型形式に対する類似 (JAPANESE)
[ 講演概要 ]
Sturm は重さ$k$, レベル$\\Gamma_1(N)$ のmod $\\ell$ 正則楕円保型形式が最初
の$(k/12)[\\Gamma_1(1):\\Gamma_1(N)]$ までの mod $\\ell$ Fourier 係数で決ま
ることを示した.

### 2011年02月10日(木)

11:00-12:00   数理科学研究科棟(駒場) 056号室

Joseph Ayoub 氏 (University of Zurich)
The motivic Galois group and periods of algebraic varieties (ENGLISH)
[ 講演概要 ]
We give a construction of the motivic Galois group of $\\Q$ and explain the conjectural link with the ring of periods of algebraic varieties. Then we introduce the ring of formal periods and explain how the conjectural link with the motivic Galois group can be realized for them.

### 2011年01月26日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室

[ 講演概要 ]
p進Gross-Zagier公式は, 楕円曲線のp進L関数の微分値をHeegner点のp進高さで記述する公式である. 楕円曲線がpで通常還元をもつときは, 20年以上前にPerrin-Riouによって証明されていた. 最近, pで超特異還元をもつときにも証明できたのでそれを紹介する. この講演では特に証明の解説に重点をおいて話したい.

### 2011年01月12日(水)

16:30-18:45   数理科学研究科棟(駒場) 056号室
Zhonghua Li 氏 (東京大学大学院数理科学研究科) 16:30-17:30
On regularized double shuffle relation for multiple zeta values (ENGLISH)
[ 講演概要 ]
Multiple zeta values(MZVs) are natural generalizations of Riemann zeta values. There are many rational relations among MZVs. It is conjectured that the regularized double shuffle relations contian all rational relations of MZVs. So other rational relations should be deduced from regularized dhouble shuffle relations. In this talk, we discuss some results on this problem. We define the gamma series accociated to elements satisfying regularized double shuffle relations and give some properties. Moreover we show that the Ohno-Zagier relations can be deduced from regularized double shuffle relations.
Dan Yasaki 氏 (North Carolina University) 17:45-18:45
Spines with View Toward Modular Forms (ENGLISH)
[ 講演概要 ]
The study of an arithmetic group is often aided by the fact that it acts naturally on a nice topological object. One can then use topological or geometric techniques to try to recover arithmetic data. For example, one often studies SL_2(Z) in terms of
its action on the upper half plane. In this talk, we will examine spines, which are the smallest" such spaces for a given arithmetic group. On overview of some known theoretical results and explicit computations will be given.

### 2010年12月22日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室

[ 講演概要 ]

「ゼータ関数の貼り合わせ」の手法を用いて加藤、Mahesh Kakde 及び

Alfred Weiss も異なる定式化の下で主予想が成立する例を構成している)。

なお、総実代数体の非可換岩澤主予想は、2010年に
Ritter-Weiss 及び Kakde によって一般の場合にも

### 2010年12月01日(水)

16:30-18:45   数理科学研究科棟(駒場) 056号室

On a problem of Matsumoto and Tamagawa concerning monodromic fullness of hyperbolic curves (JAPANESE)
[ 講演概要 ]
In this talk, we will discuss the following problem posed by Makoto Matsumoto and Akio Tamagawa concerning monodromic fullness of hyperbolic curves.

For a hyperbolic curve X over a number field, are the following three conditions equivalent?
(A) For any prime number l, X is quasi-l-monodromically full.
(B) There exists a prime number l such that X is l-monodromically full.
(C) X is l-monodromically full for all but finitely many prime numbers l.

The property of being (quasi-)monodromically full may be regarded as an analogue for hyperbolic curves of the property of not admitting complex multiplication for elliptic curves, and the above equivalence may be regarded as an analogue for hyperbolic curves of the following result concerning the Galois representation on the Tate module of an elliptic curve over a number field proven by Jean-Pierre Serre.

For an elliptic curve E over a number field, the following four conditions are equivalent:
(0) E does not admit complex multiplication.
(1) For any prime number l, the image of the l-adic Galois representation associated to E is open.
(2) There exists a prime number l such that the l-adic Galois representation associated to E is surjective.
(3) The l-adic Galois representation associated to E is surjective for all but finitely many prime numbers l.

In this talk, I will present some results concerning the above problem in the case where the given hyperbolic curve is of genus zero. In particular, I will give an example of a hyperbolic curve of type (0,4) over a number field which satisfies condition (C) but does not satisfy condition (A).
Marco Garuti 氏 (University of Padova) 17:45-18:45
Galois theory for schemes (ENGLISH)
[ 講演概要 ]
We discuss some aspects of finite group scheme actions: the Galois correspondence and the notion of Galois closure.

### 2010年11月17日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室

F_2-線形擬似乱数発生法の評価に用いる格子の簡約基底計算の高速化 (JAPANESE)
[ 講演概要 ]
(部分的に松本眞氏、斎藤睦夫氏との共同研究)

の一つとして、高次元均等分布性がしばしば用いられる。メルセンヌツイスター法
を含む二元体上の線形擬似乱数発生法に対しては、上位ビットの均等分布の次元を

L'Ecuyer-Tezuka(1993)およびTezuka(1994))。本研究では、前述の格子を用いた

(i) 冪級数成分の格子点を擬似乱数発生器の状態ベクトルで表現する、
(ii) 射影を用いてv次元簡約基底からv-1次元簡約基底を計算する、
(iii) 効率的な格子簡約アルゴリズムを適用する、
などの手法を導入し、均等分布の次元計算の高速化を提案する。この方法は、
Couture-L'Ecuyer(2000)による双対格子を用いた改良よりも計算量が少なく、計算機

### 2010年10月06日(水)

16:30-17:30   数理科学研究科棟(駒場) 117号室
いつもと教室が異なりますのでご注意ください
Hélène Esnault 氏 (Universität Duisburg-Essen)
Finite group actions on the affine space (ENGLISH)
[ 講演概要 ]
If $G$ is a finite $\\ell$-group acting on an affine space $\\A^n$ over a
finite field $K$ of cardinality prime to $\\ell$, Serre shows that there
exists a rational fixed point. We generalize this to the case where $K$ is a
henselian discretely valued field of characteristic zero with algebraically
closed residue field and with residue characteristic different from $\\ell$.
We also treat the case where the residue field is finite of cardinality $q$
such that $\\ell$ divides $q-1$. To this aim, we study group actions on weak
N\\'eron models.
(Joint work with Johannes Nicaise)

### 2010年07月07日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室

On the stable reduction of $X_0(p^4)$ (JAPANESE)

### 2010年06月16日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室
Luc Illusie 氏 (Universite de Paris-Sud)
Vanishing theorems revisited, after K.-W. Lan and J. Suh (ENGLISH)
[ 講演概要 ]
Let k be an algebraically closed field of characteristic p and X,
Y proper, smooth k-schemes. J. Suh has proved a vanishing theorem of Kollar
type for certain nef and big line bundles L on Y and morphisms f : X -> Y
having semistable reduction along a divisor with simple normal crossings. It
holds both if p = 0 and if p > 0 modulo some additional liftability mod p^2
and dimension assumptions, and generalizes vanishing theorems of Esnault-
Viehweg and of mine. I'll give an outline of the proof and sketch some
applications, due to K.-W. Lan and J. Suh, to the cohomology of certain
automorphic bundles arising from PEL type Shimura varieties.

### 2010年06月09日(水)

16:15-17:15   数理科学研究科棟(駒場) 052号室
Richard Hain 氏 (Duke大学)
Universal mixed elliptic motives (ENGLISH)
[ 講演概要 ]
This is joint work with Makoto Matsumoto. A mixed elliptic
motive is a mixed motive (MHS, Galois representation, etc) whose
weight graded quotients are Tate twists of symmetric powers of the the
motive of elliptic curve. A universal mixed elliptic motive is an
object that can be specialized to a mixed elliptic motive for any
elliptic curve and whose specialization to the nodal cubic is a mixed
Tate motive. Universal mixed elliptic motives form a tannakian
category. In this talk I will define universal mixed elliptic motives,
give some fundamental examples, and explain what we know about the
fundamental group of this category. The "geometric part" of this group
is an extension of SL_2 by a prounipotent group that is generated by
Eisenstein series and which has a family of relations for each cusp
form. Although these relations are not known, we have a very good idea
of what they are, thanks to work of Aaron Pollack, who determined
relations between the generators in a very large representation of
this group.

### 2010年06月09日(水)

17:30-18:30   数理科学研究科棟(駒場) 056号室
Fabrice Orgogozo 氏 (CNRS, École polytechnique)
エタールコホモロジーの高次順像の一様構成可能性について
(ENGLISH)
[ 講演概要 ]
Z_ℓエタールコホモロジーの捻れとF_ℓコホモロジーの超積の関係を巡り
N. Katz氏の指摘に基づいて、高次順像に於けるℓに対する

(この様な定理は以前よりガバー氏の構想に有った。)
ここでは月並みな方法で有るが、A.J.de Jong氏の定理と

(本講演は「東京パリ数論幾何セミナー」として、インターネットによる東大数理とIHESとの双方向同時中継で行います。)

### 2010年06月02日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室

On some algebraic properties of CM-types of CM-fields and their
reflex fields (JAPANESE)
[ 講演概要 ]
Shimura and Taniyama proved in their theory of complex
multiplication that the moduli of abelian varieties of a CM-type and their
torsion points generate an abelian extension, not of the field of complex
multiplication, but of a reflex field of the field. In this talk, I
introduce some algebraic properties of CM-types, half norm maps that might
shed new light on reflex fields.

For a CM-field $K$ and its Galois closure $K^c$ over the rational field $Q$,
there is a canonical embedding of $Gal (K^c/Q)$ into $(Z/2Z)^n \\rtimes S_n$.
Using properties of the embedding, a set of CM-types $\\Phi$ of $K$ and their
dual CM-types $(K, \\Phi)$ is equipped with a combinatorial structure. This
makes it much easier to handle a whole set of CM-types than an individual
CM-type.

I present a theorem that shows the combinatorial structure of the dual
CM-types is isomorphic to that of a Pfister form.

### 2010年05月12日(水)

17:30-18:30   数理科学研究科棟(駒場) 056号室

Differences between
Galois representations in outer-automorphisms
of the fundamental groups and those in automorphisms, implied by
topology of moduli spaces (ENGLISH)
[ 講演概要 ]
Fix a prime l. Let C be a proper smooth geometrically connected curve over a number ﬁeld K, and x be its closed point. Let Π denote the pro-l completion of the geometric fundamental group of C with geometric base point over x. We have two non-abelian Galois representations:

ρA : Galk(x) → Aut(Π),ρO : GalK → Out(Π).

Our question is: in the natural inclusion Ker(ρA) ⊂ Ker(ρO) ∩ Galk(x), whether the equality holds or not. Theorem: Assume that g ≥ 3, l divides 2g -2. Then, there are inﬁnitely many pairs (C, K) with the following property. If l does not divide the extension degree [k(x): K], then Ker(ρA) = (Ker(ρO) ∩ Galk(x)) holds.

This is in contrast to the case of the projective line minus three points and its canonical tangential base points, where the equality holds (a result of Deligne and Ihara).

There are two ingredients in the proof: (1) Galois representations often contain the image of the geometric monodromy (namely, the mapping class group) [M-Tamagawa 2000] (2) A topological result [S. Morita 98] [Hain-Reed 2000] on the cohomological obstruction of lifting the outer action of the mapping class group to automorphisms.

(This lecture is held as `Arithmetic Geometry Seminar Tokyo-Paris' and it is transmitted to IHES by the internet.)

### 2010年04月14日(水)

17:30-18:30   数理科学研究科棟(駒場) 056号室
Gerard Laumon 氏 (CNRS, Universite Paris XI - Orsay)
The cohomological weighted fundamental lemma
[ 講演概要 ]
Using the Hitchin fibration, Ngo Bao Chau has proved the Langlands-Shelstad fundamental lemma. In a joint work with Pierre-Henri Chaudouard, we have extended Ngo's proof to obtain the weighted fundamental lemma which had been conjectured by Arthur. In the talk, I would like to present our main cohomological result.

(本講演は「東京パリ数論幾何セミナー」として、インターネットによる東大数理とIHESとの双方向同時中継で行います。)

### 2009年11月18日(水)

16:30-18:45   数理科学研究科棟(駒場) 056号室

Elementary computation of ramified component of the Jacobi sum
[ 講演概要 ]
R. Coleman and W. McCallum calculated the Jacobi sum Hecke characters using their computation of the stable reduction of the Fermat curve in 1988. In my talk, we give an elementary proof of the main result of them without using rigid geometry or the stable model of the Fermat curve.
Christopher Deninger 氏 (Universität Münster) 17:45-18:45
P-divisible groups and the p-adic Corona problem

### 2009年10月21日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室
Bernard Le Stum 氏 (Université de Rennes 1)
The local Simpson correspondence in positive characteristic
[ 講演概要 ]
A Simpson correspondance should relate Higgs bundles to differential modules (or local systems). We stick here to positive characteristic and recall some old and recent results : Cartier isomorphism, Van der Put's classification, Kaneda's theorem and Ogus-Vologodsky local theory. We'll try to explain how the notion of Azumaya algebra is a convenient tool to unify these results. Our main theorem is the equivalence between quasi-nilpotent differential modules of level m and quasi-nilpotent Higgs Bundles (depending on a lifting of Frobenius mod p-squared). This result is a direct generalization of the previous ones. The main point is to understand the Azumaya nature of the ring of differential operators of level m. Following Berthelot, we actually use the dual theory and study the partial divided power neighborhood of the diagonal.

### 2009年10月07日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室
Ahmed Abbes 氏 (Université de Rennes 1)
On GAGA theorems for the rigide-étale topology
[ 講演概要 ]
Last year, I finished my course in Todai on "Rigide Geometry following M. Raynaud" by stating a GAGA theorem for the rigide-étale topology, due to Gabber and Fujiwara. I will give a new proof of this theorem, inspired by another theorem of Gabber, namely the Affine analog of the proper base change theorem.

### 2009年09月14日(月)

11:00-12:00   数理科学研究科棟(駒場) 123号室
いつもと、曜日、時間、教室が違います。

Dinakar Ramakrishnan 氏 (カリフォルニア工科大学)
Modular forms and Calabi-Yau varieties

### 2009年08月07日(金)

16:30-17:30   数理科学研究科棟(駒場) 117号室
いつもと曜日が違います。
Fabien Trihan 氏 (Nottingham大学)
On the $p$-parity conjecture in the function field case
[ 講演概要 ]
Let $F$ be a function field in one variable with field of constant a finite field of characteristic $p>0$. Let $E/F$ be an elliptic curve over $F$. We show that the order of the Hasse-Weil $L$-function of $E/F$ at $s=1$ and the corank of the $p$-Selmer group of $E/F$ have the same parity (joint work with C. Wuthrich).

### 2009年06月24日(水)

16:30-18:45   数理科学研究科棟(駒場) 056号室
Vincent Maillot 氏 (Paris第7大学) 16:30-17:30
New algebraicity results for analytic torsion
Richard Hain 氏 (Duke大学) 17:45-18:45
On the Section Conjecture for the universal curve over function fields

### 2009年06月10日(水)

16:30-18:30   数理科学研究科棟(駒場) 056号室
Bruno Kahn 氏 (Paris第7大学)
On the classifying space of a linear algebraic group

### 2009年06月03日(水)

16:30-18:30   数理科学研究科棟(駒場) 056号室
Bruno Kahn 氏 (Paris第7大学)