## 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

### 2013年10月30日(水)

16:40-17:40   数理科学研究科棟(駒場) 002号室
いつもと場所が異なりますのでご注意ください
Pierre Charollois 氏 (パリ第6大学)
Explicit integral cocycles on GLn and special values of p-adic partial zeta functions (ENGLISH)
[ 講演概要 ]
Building on earlier work by Sczech, we contruct an explicit integral valued cocycle on GLn(Z).
It allows for the detailed analysis of the order of vanishing and of the special value at s=0 of the p-adic partial zeta functions introduced by Pi. Cassou-Noguès and Deligne-Ribet. In particular we recover a result of Wiles (1990) on Gross conjecture.
Another construction, now based on Shintani's method, is shown to lead to a cohomologous cocycle. This is joint work with S. Dasgupta and M. Greenberg.

### 2013年10月16日(水)

17:30-18:30   数理科学研究科棟(駒場) 056号室
Peter Scholze 氏 (ボン大学)
Shimura varieties with infinite level, and torsion in the cohomology of locally symmetric spaces (ENGLISH)
[ 講演概要 ]
We will discuss the p-adic geometry of Shimura varieties with infinite level at p: They are perfectoid spaces, and there is a new period map defined at infinite level. As an application, we will discuss some results on torsion in the cohomology of locally symmetric spaces, and in particular the existence of Galois representations in this setup.

(本講演は「東京北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, Morningside Center of MathematicsとIHESの双方向同時中継で行います.)

### 2013年07月24日(水)

16:40-17:40   数理科学研究科棟(駒場) 056号室

The determinant of a double covering of the projective space of even dimension and the discriminant of the branch locus (JAPANESE)

### 2013年07月10日(水)

17:00-18:00   数理科学研究科棟(駒場) 056号室

On ramification filtration of local fields of equal characteristic (JAPANESE)

### 2013年07月03日(水)

16:40-17:40   数理科学研究科棟(駒場) 056号室

A general formula for the discriminant of polynomials over $¥mathbb{F}_2$ determining the parity of the number of prime factors
(JAPANESE)
[ 講演概要 ]
In order to find irreducible polynomials over $\\mathbb{F}_2$ efficiently, the method using Swan's theorem is known. Swan's theorem determines the parity of the numberof irreducible factors of a polynomial $f$ over $\\mathbb{F}_2$ with no repeated root, by using the discriminant ${\\rm D}(\\tilde{f})\\pmod 8$, where $\\tilde{f}$ is a monic polynomial over $\\mathbb{Z}_2$ such that $\\tilde{f}=f\\pmod 2$. In the lecture, we will give the formula for the discriminant ${\\rm D}(\\tilde{f}) \\pmod 8$ for a polynomial $f$ over $\\mathbb{F}_2$ with no repeated root. By applying this formula to various types of polynomials, we shall get the parity of the number of irreducible factors of them.

### 2013年06月26日(水)

16:40-17:40   数理科学研究科棟(駒場) 056号室

An explicit construction of point sets with large minimum Dick weight (JAPANESE)
[ 講演概要 ]
Walsh figure of merit WAFOM($P$) is a quality measure of point sets $P$ for quasi-Monte Carlo integration constructed by a digital net method. WAFOM($P$) is bounded by the minimum Dick weight of $P^¥perp$, where the Dick weight is a generalization of Hamming weight. In this talk, we give an explicit construction of point sets with large minimum Dick weight using Niederreiter-Xing sequences and Dick's interleaving construction. These point sets are also examples of low-WAFOM point sets.

### 2013年06月19日(水)

16:40-17:40   数理科学研究科棟(駒場) 056号室

A p-adic exponential map for the Picard group and its application to curves (JAPANESE)
[ 講演概要 ]
Let $\\mathcal{X}$ be a proper flat scheme over a complete discrete valuation ring $O_k$ of characteristic $(0,p)$. We define an exponential map from a subgroup of the first cohomology group of $O_¥mathcal{X}$ to the Picard group of $\\mathcal{X}$, mimicking the classical construction in complex geometry. This exponential map can be applied to prove a surjectivity property concerning the Albanese variety $Alb_{X}$ of a smooth variety $X$ over $k$.

### 2013年06月12日(水)

17:30-18:30   数理科学研究科棟(駒場) 056号室
Xinyi Yuan 氏 (University of California, Berkeley)
Hodge index theorem for adelic line bundles (ENGLISH)
[ 講演概要 ]
The Hodge index theorem of Faltings and Hriljac asserts that the Neron-Tate height pairing on a projective curve over a number field is equal to certain intersection pairing in the setting of Arakelov geometry. In the talk, I will present an extension of the result to adelic line bundles on higher dimensional varieties over finitely generated fields. Then we will talk about its relation to the non-archimedean Calabi-Yau theorem and the its application to algebraic dynamics. This is a joint work with Shou-Wu Zhang.

(本講演は「東京北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, Morningside Center of MathematicsとIHESの双方向同時中継で行います.)

### 2013年05月29日(水)

16:40-17:40   数理科学研究科棟(駒場) 002号室

On logarithmic nonabelian Hodge theory of higher level in characteristic p (JAPANESE)
[ 講演概要 ]
Ogus and Vologodsky studied a positive characteristic analogue of Simpson’s nonanelian Hodge theory over the complex number field. Now most part of their theory has been generalized to the case of log schemes by Schepler. In this talk, we generalize the global Cartier transform, which is one of the main theorem in nonabelian Hodge theory in positive characteristic, to the case of log schemes and of higher level. This can be regarded as a higher level version of a result of Schepler.

### 2013年05月15日(水)

16:40-17:40   数理科学研究科棟(駒場) 056号室

Special values of zeta functions of singular varieties over finite fields via higher chow groups (JAPANESE)

### 2013年04月24日(水)

17:30-18:30   数理科学研究科棟(駒場) 056号室

Good reduction of ramified affinoids in the Lubin-Tate perfectoid space (ENGLISH)
[ 講演概要 ]
Recently, Weinstein finds some affinoids in the Lubin-Tate perfectoid space and computes their reduction in equal characteristic case. The cohomology of the reduction realizes the local Langlands correspondence for some representations of GL_h, which are unramified in some sense. In this talk, we introduce other affinoids in the Lubin-Tate perfectoid space in equal characteristic case, whose reduction realizes "ramified" representations of conductor exponent h+1. We call them ramified affinoids. We study the cohomology of the reduction and its relation with the local Langlands correspondence. This is a joint work with Takahiro Tsushima.

(本講演は「東京北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, Morningside Center of MathematicsとIHESの双方向同時中継で行います.)

### 2013年04月10日(水)

17:30-18:30   数理科学研究科棟(駒場) 056号室
Deepam Patel 氏 (University of Amsterdam)
Motivic structure on higher homotopy of non-nilpotent spaces (ENGLISH)
[ 講演概要 ]
In his fundamental paper on the projective line minus three points, Deligne constructed certain extensions of mixed Tate motives arising from the fundamental group of the projective line minus three points. Since then, motivic structures on homotopy groups have been studied by many authors. In this talk, we will construct a motivic structure on the (nilpotent completion of) n-th homotopy group of P^{n} minus n+2 hyperplanes in general position.

(本講演は「東京北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, Morningside Center of MathematicsとIHESの双方向同時中継で行います.)

### 2013年01月16日(水)

18:00-19:00   数理科学研究科棟(駒場) 002号室

On differential modules associated to de Rham representations in the imperfect residue field case (ENGLISH)
[ 講演概要 ]
Let K be a CDVF of mixed characteristic (0,p) and G the absolute Galois group of K. When the residue field of K is perfect, Laurent Berger constructed a p-adic differential equation N_dR(V) for any de Rham representation V of G. In this talk, we will generalize his construction when the residue field of K is not perfect. We also explain some ramification properties of our N_dR, which are due to Adriano Marmora in the perfect residue field case.

(本講演は「東京北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, Morningside Center of MathematicsとIHESの双方向同時中継で行います.)

### 2012年12月19日(水)

16:40-17:40   数理科学研究科棟(駒場) 056号室

A generalization of Kato's local epsilon conjecture for
(φ, Γ)-modules over the Robba ring (JAPANESE)
[ 講演概要 ]
In his preprint “Lectures on the approach to Iwasawa theory of Hasse-Weil L-functions via B_dR, Part II ", Kazuya Kato proposed a conjecture called local epsilon conjecture. This conjecture predicts that the determinant of the Galois cohomology of a family of p-adic Galois representations has a canonical base whose specializations at de Rham points can be characterized by using Bloch-Kato exponential, L-factors and Deligne-Langlands epsilon constants of the associated Weil-Deligne representations.
In my talk, I generalize his conjecture for families of (φ, Γ)-modules over the Robba ring, and prove a part of this conjecture in the trianguline case. The two key ingredients are the recent result of Kedlaya-Pottharst-Xiao on the ﬁniteness of cohomologies of (φ, Γ)-modules and my result on Bloch-Kato exponential map for (φ, Γ)-modules.

### 2012年12月12日(水)

18:00-19:00   数理科学研究科棟(駒場) 002号室
François Charles 氏 (CNRS & Université de Rennes 1)
The Tate conjecture for K3 surfaces and holomorphic symplectic varieties over finite fields (ENGLISH)
[ 講演概要 ]
We prove the Tate conjecture for divisors on reductions of holomorphic symplectic varieties over finite fields -- with some restrictions on the characteristic of the base field. We will be concerned mostly with the supersingular case. As a special case, we prove the Tate conjecture, also known as Artin's conjecture in our case, for K3 surfaces over finite fields of characteristic at least 5 and for codimension 2 cycles on cubic fourfolds.

(本講演は「東京北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, Morningside Center of MathematicsとIHESの双方向同時中継で行います.)

### 2012年11月14日(水)

18:00-19:00   数理科学研究科棟(駒場) 002号室
いつもと時間・場所が異なりますのでご注意ください.
Pierre Berthelot 氏 (Université de Rennes 1)
De Rham-Witt complexes with coefficients and rigid cohomology
(ENGLISH)
[ 講演概要 ]
For a smooth scheme over a perfect field of characteristic p, we will explain a generalization of the classical comparison theorem between crystalline cohomology and de Rham-Witt cohomology to the case of cohomologies with coefficients in a p-torsion free crystal. This provides in particular a description of the rigid cohomology of a proper singular scheme in terms of a de Rham-Witt complex built from a closed immersion into a smooth scheme.
(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.)

### 2012年07月18日(水)

16:40-17:40   数理科学研究科棟(駒場) 056号室
Shane Kelly 氏 (Australian National University)
Voevodsky motives and a theorem of Gabber (ENGLISH)
[ 講演概要 ]
The assumption that the base field satisfies resolution of singularities litters Voevodsky's work on motives. While we don't have resolution of singularities in positive characteristic p, there is a theorem of Gabber on alterations which may be used as a substitute if we are willing to work with Z[1/p] coefficients. We will discuss how this theorem of Gabber may be applied in the context of Voevodsky's work and mention some consequences.

### 2012年07月04日(水)

16:40-17:40   数理科学研究科棟(駒場) 056号室
Patrick Forré 氏 (東京大学数理科学研究科)
A cohomological Hasse principle of varieties over higher local fields and applications to higher dimensional class field theory (ENGLISH)
[ 講演概要 ]
In this talk I will give an overview of the necessary tools for a description of the class field theory of varieties over higher local fields developed by sevaral mathematicians. On this I will motivate the importance of the proposal and verification of a cohomological Hasse principle for varieties over higher local fields, a generalization of Kato's conjectures, and sketch the recent progress on this.

### 2012年06月20日(水)

16:40-17:40   数理科学研究科棟(駒場) 056号室

On the reduction modulo p of representations of a quaternion
division algebra over a p-adic field (JAPANESE)
[ 講演概要 ]
The p-adic Langlands correspondence and the mod p Langlands correspondence for GL_2(Q_p) are known to be compatible with the reduction modulo p in many cases.
In this talk, we examine whether a similar compatibility exists for the composition of the local Langlands correspondence and the local Jacquet-Langlands correspondence.
The simplest case has already been treated by Vign¥'eras. We deal with more cases.

### 2012年06月13日(水)

16:40-17:40   数理科学研究科棟(駒場) 056号室

Singular homologies of non-Archimedean analytic spaces and integrals along cycles (JAPANESE)
[ 講演概要 ]
Berkovichの非アルキメデス的解析空間に新たな特異ホモロジーを定義し、そのホモロジーにおけるサイクルに沿った正則微分形式の積分という新たな概念を考える。Tate曲線においては標準的な体積形式dz/zの積分により周期が得られることを確かめる。

### 2012年05月30日(水)

16:40-17:40   数理科学研究科棟(駒場) 056号室
Valentina Di Proietto 氏 (東京大学数理科学研究科)
Kernel of the monodromy operator for semistable curves (ENGLISH)
[ 講演概要 ]
For a semistable curve, we study the action of the monodromy operator on the first log-crystalline cohomology group. In particular we examine the relation between the kernel of the monodromy operator and the first rigid cohomology group, in the case of trivial coefficients, giving a new proof of a theorem of B. Chiarellotto and in the case of certain unipotent F-isocrystals as coefficients.
This is a joint work in progress with B. Chiarellotto, R. Coleman and A. Iovita.

### 2012年05月23日(水)

16:40-17:40   数理科学研究科棟(駒場) 056号室

Simply connected elliptic surfaces (JAPANESE)
[ 講演概要 ]
We characterize simply connected elliptic surfaces by their singular fibers in any characteristic case. To this end, we study orbifolds of curves, local canonical bundle formula, and resolutions of multiple fibers. The result was known for the complex analytic case. Our method can be applied to the rigid analytic case.

### 2012年05月16日(水)

16:40-17:40   数理科学研究科棟(駒場) 002号室

On uniform bound of the maximal subgroup of the inertia group acting unipotently on $¥ell$-adic cohomology (JAPANESE)
[ 講演概要 ]
For a smooth projective variety over a local field,
the action of the inertia group on the $¥ell$-adic cohomology group is
unipotent if it is restricted to some open subgroup.
In this talk, we give a uniform bound of the index of the maximal open
subgroup satisfying this property.
This bound depends only on the Betti numbers of $X$ and certain Chern
numbers depending on a projective embedding of $X$.

### 2012年04月18日(水)

16:40-17:40   数理科学研究科棟(駒場) 056号室
Alan Lauder 氏 (University of Oxford)
Explicit constructions of rational points on elliptic curves (ENGLISH)
[ 講演概要 ]
I will present an algorithm for computing certain special
values of p-adic L-functions, and discuss an application to
the efficient construction of rational points on elliptic curves.

### 2012年04月11日(水)

17:30-18:30   数理科学研究科棟(駒場) 056号室
Damian Rossler 氏 (CNRS, Universite de Toulouse)
Around the Mordell-Lang conjecture in positive characteristic (ENGLISH)
[ 講演概要 ]
Let V be a subvariety of an abelian variety A over C and let G\\subseteq A(C) be a subgroup. The classical Mordell-Lang conjecture predicts that if V is of general type and G\\otimesQ is finite dimensional, then V\\cap G is not Zariski dense in V. This statement contains the Mordell conjecture as well as the Manin-Mumford conjecture (for curves). The positive characteristic analog of the Mordell-Lang conjecture makes sense, when A is supposed to have no subquotient, which is defined over a finite field. This positive characteristic analog was proven in 1996 by E. Hrushovski using model-theoretic methods. We shall discuss the prehistory and context of this proof. We shall also discuss the proof (due to the speaker) of the fact that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture (whereas this seems far from true in characteristic 0).

(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。)