## 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

### 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)

### 2009年05月27日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室
Gombodorj Bayarmagnai 氏 (東京大学大学院数理科学研究科)
The (g,K)-module structure of principal series and related Whittaker functions of SU(2,2)

### 2009年05月20日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室

Generalized Whittaker functions for degenerate principal series of GL(4,R)

### 2009年05月13日(水)

16:30-18:45   数理科学研究科棟(駒場) 056号室

A counterexample of Bloch-Kato conjecture over a local field and infinite torsion in algebraic cycles of codimension two

### 2009年01月28日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室
Pierre Colmez 氏 (École polytechnique)
On the p-adic local Langlands correspondence

### 2008年12月03日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室

Mean-periodicity and analytic properties of zeta-functions
[ 講演概要 ]
Mean-periodicityというのは周期性の概念のひとつの一般化である。最近、I. Fesenko, G. Ricottaとの共同研究により、数論的スキームのゼータ関数を含むある複素関数のクラスと、mean-periodicityとの関連性が新しく見出された。
これはHecke-Weilによる, 解析接続と関数等式を持つDirichlet級数と保型形式との対応の一つの拡張ともみなせる. この背景には, I. Fesenkoの高次元アデール上のゼータ積分の理論があり、数論的スキームのHasseゼータ関数の解析接続を高次元アデール上の調和解析から導こうというプログラムの一環となっている。
この講演ではそのような背景にも若干触れた上、ゼータ関数の解析的性質とmean-periodicityの関連、特に解析接続と関数等式との関連について解説する。

### 2008年11月26日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室

Lang's Observation in Diophantine Problems
[ 講演概要 ]
In 1964, Serge Lang suggested the following problem, which reads now as follows:
Let $E$ be an elliptic curve defined over a number field $K$, and $\\varphi$ be a rational function on $E$. Then, for every point $P\\in E(K)$ where $\\varphi$ does not vanish at $P$, the logarithms of a norm of $\\varphi(P)$ is at worst linear in the logarithms of the Neron-Tate height of the point $P$.
We give a simultaneous Diophantine approximation for linear forms in elliptic logarithms which actually implies this conjecture. We also present Lang's observations in Diophantine problems.

### 2008年11月19日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室
Olivier Fouquet 氏 (大阪大学)
Dihedral Iwasawa theory of ordinary modular forms
[ 講演概要 ]
According to Hida theory, the Galois representation attached to a nearly-ordinary Hilbert eigencuspform belongs to a p-adic analytic family of Galois representations parametrized by varying weights. After restricting it to the absolute Galois group of a quadratic totally complex extension, it also belongs to a p-adic family coming from classical dihedral Iwasawa theory. We will explain the proofs of part of the main conjecture in Iwasawa theory in these situations, i.e divisibilities of characteristic ideals when equalities are actually expected.

### 2008年10月29日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室
Daniel Caro 氏 (Université de Caen)
Overholonomicity of overconvergence $F$-isocrystals on smooth varieties
[ 講演概要 ]
Let $¥mathcal{V}$ be a complete discrete valuation ring
of characteristic $0$, with perfect residue field $k$ of
characteristic $p>0$. In order to construct $p$-adic coefficients
over $k$-varieties, Berthelot introduced the theory of
overconvergent $F$-isocrystals, i.e overconvergent isocrystals with
Frobenius structure. Moreover, to get a $p$-adic cohomology over
$k$-varieties stable under cohomological operations, Berthelot built
the theory of arithmetic $F$-$¥mathcal{D}$-modules. In this talk,
after recalling some elements of these theories, we introduce the
notion of overholonomicity with is a property as stable as the
holonomicity in the classical theory of $¥mathcal{D}$-modules. The
goal of the talk is to prove the overholonomicity of arithmetic
$¥mathcal{D}$-modules associated to overconvergent $F$-isocrystals
over smooth $k$-varieties. In the proof we need Christol's transfert
theorem, a comparison theorem between relative log rigid cohomology
and relative rigid cohomology and last but not least Kedlaya's
semistable reduction theorem. This is a joint work with Nobuo
Tsuzuki.

### 2008年10月22日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室
10月から教室が056号室に変更になります.
Pierre Parent 氏 (Universite Bordeaux 1)
Serre's uniformity in the split Cartan case
[ 講演概要 ]
We show that, for large enough prime number p, the modular curve
X_{split}(p) has no other point with values in Q than CM points and the rational cusp. This gives a partial answer to an old question of J.-P. Serre concerning the uniform surjectivity of Galois representations associated to torsion points on elliptic curves without complex multiplication.
(Joint work with Yuri Bilu.)

### 2008年09月29日(月)

16:30-17:30   数理科学研究科棟(駒場) 117号室
いつもと曜日が異なりますのでご注意下さい.
Christopher Deninger 氏 (Munster大学)
A determinant for p-adic group algebras
[ 講演概要 ]
For a discrete countable group G there is a classical determinant on the units of the L^1-convolution algebra of G. It is defined using functional analysis and can be used for example to calculate the entropy of certain G-actions. We will discuss a p-adic analogue of this theory. Instead of functional analysis the definition of the p-adic determinant uses algebraic K-theory. It has an application to the study of the p-adic distribution of periodic G-orbits in certain G-action.

### 2008年08月27日(水)

16:30-17:30   数理科学研究科棟(駒場) 117号室
Don Zagier 氏 (Max Planck研究所)
$q$-series and modularity

### 2008年08月01日(金)

13:00-18:00   数理科学研究科棟(駒場) 002号室
4講演あります.また,いつもと曜日,時間,場所が異なります.
ご注意下さい.

Olivier Brinon 氏 (Paris北大学) 13:00-14:00
B_dR-representations and Higgs bundles
Henrik Russell 氏 (Duisburg-Essen大学) 14:15-15:15
Generalized Albanese and duality
Thomas Geisser 氏 (南California大学) 15:45-16:45
Negative K-theory, homotopy invariance and regularity
[ 講演概要 ]
The topic of my talk are two classical conjectures in K-theory:
Weibel's conjecture states that a scheme of dimension d
has no K-groups below degree -d, and Vorst's conjecture
states that homotopy invariance of the K-theory of rings
implies that the ring must be regular.
I will give an easy introduction to the conjectures, and discuss
recent progress.
Fabien Trihan 氏 (Nottingham大学) 17:00-18:00
On Iwasawa theory for abelian varieties over function fields of positive characteristic