## 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

### 2017年05月10日(水)

17:00-18:00   数理科学研究科棟(駒場) 056号室

Wild ramification and restrictions to curves (JAPANESE)
[ 講演概要 ]
スキーム上のエタール層の暴分岐がすべての曲線への制限のArtin導手で決まるかどうかを調べ、特異点解消を仮定するとそれが正しいこと、特に、スキームが2次元の場合には正しいことを示した。
またその帰結として、(次元に関する仮定なしに)体上の多様体のエタール層のEuler－Poincare標数や、局所体上の多様体のエタール層から定まるGalois表現のSwan導手の交代和もすべての"曲線"への制限のArtin導手で決まるという結果を得た。

### 2017年04月12日(水)

17:00-18:00   数理科学研究科棟(駒場) 056号室

A conjecture of Gross-Prasad and Rallis for metaplectic groups (JAPANESE)
[ 講演概要 ]
p-進簡約代数群の既約スムース表現が generic であるとは、それが Whittaker 模型を持つ時に言う。
Whittaker 模型の一意性により、generic 表現は表現論及び数論の両分野で多くの応用を持つ。

Gross-Prasad は Rallis に触発されて、generic 表現に対応する L-パラメーターの判定法を予想した。
これを Gross-Prasad と Rallis の予想 (GPR) という。

### 2017年04月11日(火)

17:30-18:30   数理科学研究科棟(駒場) 117号室
Peter Scholze 氏 (University of Bonn)
The geometric Satake equivalence in mixed characteristic (ENGLISH)
[ 講演概要 ]
In order to apply V. Lafforgue's ideas to the study of representations of p-adic groups, one needs a version of the geometric Satake equivalence in that setting. For the affine Grassmannian defined using the Witt vectors, this has been proven by Zhu. However, one actually needs a version for the affine Grassmannian defined using Fontaine's ring B_dR, and related results on the Beilinson-Drinfeld Grassmannian over a self-product of Spa Q_p. These objects exist as diamonds, and in particular one can make sense of the fusion product in this situation; this is a priori surprising, as it entails colliding two distinct points of Spec Z. The focus of the talk will be on the geometry of the fusion product, and an analogue of the technically crucial ULA (Universally Locally Acyclic) condition that works in this non-algebraic setting.

### 2017年03月30日(木)

16:40-17:40   数理科学研究科棟(駒場) 056号室
いつもと曜日が異なりますのでご注意ください．
Haoyu Hu 氏 (東京大学数理科学研究科)
Logarithmic ramifications via pull-back to curves (English)
[ 講演概要 ]
Let X be a smooth variety over a perfect field of characteristic p>0, D a strict normal crossing divisor of X, U the complement of D in X, j:U—>X the canonical injection, and F a locally constant and constructible sheaf of F_l-modules on U (l is a prime number different from p). Using Abbes and Saito’s logarithmic ramification theory, we define a Swan divisor SW(j_!F), which supported on D. Let i:C-->X be a quasi-finite morphism from a smooth curve C to X. Following T. Saito’s idea, we compare the pull-back of SW(j_!F) to C with the Swan divisor of the pull-back of j_!F to C. It answers an expectation of Esnault and Kerz and generalizes the same result of Barrientos for rank 1 sheaves. As an application, we obtain a lower semi-continuity property for Swan divisors of an l-adic sheaf on a smooth fibration, which gives a generalization of Deligne and Laumon’s lower semi-continuity property of Swan conductors of l-adic sheaves on relative curves to higher relative dimensions. This application is a supplement of the semi-continuity of total dimension of vanishing cycles due to T. Saito and the lower semi-continuity of total dimension divisors due to myself and E. Yang.

### 2017年01月11日(水)

18:00-19:00   数理科学研究科棟(駒場) 056号室
Lei Fu 氏 (Tsinghua University)
Deformation and rigidity of $\ell$-adic sheaves (English)
[ 講演概要 ]
Let $X$ be a smooth connected algebraic curve over an algebraically closed field, let $S$ be a finite closed subset in $X$, and let $F_0$ be a lisse $\ell$-torsion sheaf on $X-S$. We study the deformation of $F_0$. The universal deformation space is a formal scheme. Its generic fiber has a rigid analytic space structure. By studying this rigid analytic space, we prove a conjecture of Katz which says that if a lisse $\overline{Q}_\ell$-sheaf $F$ is irreducible and physically rigid, then it is cohomologically rigid in the sense that $\chi(X,j_*End(F))=2$, where $j:X-S\to X$ is the open immersion.

(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.)

### 2016年12月14日(水)

18:00-19:00   数理科学研究科棟(駒場) 056号室
Luc Illusie 氏 (Université Paris-Sud)
On vanishing cycles and duality, after A. Beilinson (English)
[ 講演概要 ]
It was proved by Gabber in the early 1980's that $R\Psi$ commutes with duality, and that $R\Phi$ preserves perversity up to shift. It had been in the folklore since then that this last result was in fact a consequence of a finer one, namely the compatibility of $R\Phi$ with duality. In this talk I'll give a proof of this, using a method explained to me by A. Beilinson.

(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.)

### 2016年11月09日(水)

18:00-19:00   数理科学研究科棟(駒場) 056号室
Emmanuel Ullmo 氏 (Institut des Hautes Études Scientifiques)
Flows on Abelian Varieties and Shimura Varieties (English)
[ 講演概要 ]
I will discuss several questions and some results about algebraic flows, o-minimal flows and holomorphic flows on abelian varieties and Shimura varieties.

（本講演は東京北京パリ数論幾何セミナー」として, The Second Sino-French Conference in Arithmetic Geometry, Sanyaより中継します．インターネットによる東大数理，Morningside Center of Mathematics，IHES及びSanyaの双方向同時中継を行います.）

### 2016年11月02日(水)

18:00-19:00   数理科学研究科棟(駒場) 056号室
Yves André 氏 (CNRS, Institut de Mathématiques de Jussieu)
Direct summand conjecture and perfectoid Abhyankar lemma: an overview (English)
[ 講演概要 ]
According to Hochster's direct summand conjecture (1973), a regular ring R is a direct summand, as an R-module, of every finite extension ring. We shall outline our recent proof which relies on perfectoid techniques. Similar arguments also establish the existence of big Cohen-Macaulay algebras for complete local domains of mixed characteristics.

(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.)

### 2016年10月12日(水)

17:30-18:30   数理科学研究科棟(駒場) 056号室
Uwe Jannsen 氏 (Universität Regensburg, 東京大学数理科学研究科)
Filtered de Rham Witt complexes and wildly ramified higher class field theory over finite fields (joint work with Shuji Saito and Yigeng Zhao) (English)
[ 講演概要 ]
We will consider abelian coverings of smooth projective varieties over finite fields which are wildly ramified along a divisor D with normal crossings, and will describe the corresponding abelianized fundamental group via modified logarithmic de Rham-Witt sheaves.

(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.)

### 2016年06月08日(水)

16:00-18:30   数理科学研究科棟(駒場) 16:00-17:00は002, 17:30-18:30は056号室
Bruno Kahn 氏 (Institut de mathématiques de Jussieu-Paris Rive Gauche) 16:00-17:00
Torsion order of smooth projective surfaces (English)
[ 講演概要 ]
To a smooth projective variety $X$ whose Chow group of $0$-cycles is $\mathbb{Q}$-universally trivial one can associate its torsion order ${\mathrm{Tor}}(X)$, the smallest multiple of the diagonal appearing in a cycle-theoretic decomposition à la Bloch-Srinivas. We show that ${\mathrm{Tor}}(X)$ is the exponent of the torsion in the Néron-Severi-group of $X$ when $X$ is a surface over an algebraically closed field $k$, up to a power of the exponential characteristic of $k$.
Xu Shen 氏 (Morningside Center of Mathematics) 17:30-18:30
Local and global geometric structures of perfectoid Shimura varieties (English)
[ 講演概要 ]
In this talk, we will investigate some geometric structural properties of perfectoid Shimura varieties of abelian type. In the global part, we will construct some minimal and toroidal type compactifications for these spaces, and we will describe explicitly the degeneration of Hodge-Tate period map at the boundaries. In the local part, we will show that each Newton stratum of these perfectoid Shimura varieties can be described by the related (generalized) Rapoport-Zink space and Igusa variety. As a consequence of our local and global constructions, we can compute the stalks of the relative cohomology under the Hodge-Tate period map of the intersection complex (on the minimal compactification), in terms of cohomology of Igusa varieties at the boundary with truncated coefficients.

(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.今回は北京からの中継です．)

### 2016年05月18日(水)

17:00-18:00   数理科学研究科棟(駒場) 056号室

A consequence of Greenberg's generalized conjecture on Iwasawa invariants of Z_p-extensions (Japanese)

### 2016年05月11日(水)

17:30-18:30   数理科学研究科棟(駒場) 056号室
Wiesława Nizioł 氏 (CNRS & ENS de Lyon)
Syntomic complexes and p-adic nearby cycles (English)
[ 講演概要 ]
I will present a proof of a comparison isomorphism, up to some universal constants, between truncated sheaves of p-adic nearby cycles and syntomic cohomology sheaves on semistable schemes over a mixed characteristic local rings. This generalizes the comparison results of Kato, Kurihara, and Tsuji for small Tate twists (where no constants are necessary) as well as the comparison result of Tsuji that holds over the algebraic closure of the field. This is a joint work with Pierre Colmez.

(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.今回はパリからの中継です．)

### 2016年04月27日(水)

16:30-17:30   数理科学研究科棟(駒場) 056号室

On the endoscopic lifting of simple supercuspidal representations (Japanese)

### 2016年04月20日(水)

17:00-18:00   数理科学研究科棟(駒場) 056号室

On periodicity of geodesic continued fractions (Japanese)

### 2016年04月13日(水)

17:30-18:30   数理科学研究科棟(駒場) 056号室

Semisimplicity of geometric monodromy on etale cohomology (joint work with Anna Cadoret and Chun Yin Hui)

(English)
[ 講演概要 ]
Let K be a function field over an algebraically closed field of characteritic p \geq 0, X a proper smooth K-scheme, and l a prime distinct from p. Deligne proved that the Q_l-coefficient etale cohomology groups of the geometric fiber of X --> K are always semisimple as G_K-modules. In this talk, we consider a similar problem for the F_l-coefficient etale cohomology groups. Among other things, we show that if p=0 (resp. in general), they are semisimple for all but finitely many l's (resp. for all l's in a set of density 1).

(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.)

### 2016年03月29日(火)

17:30-18:30   数理科学研究科棟(駒場) 002号室

Matthew Morrow 氏 (Universität Bonn)
Motivic cohomology of formal schemes in characteristic p
(English)
[ 講演概要 ]
The logarithmic Hodge-Witt sheaves of Illusie, Milne, Kato, et al. of a smooth variety in characteristic p provide a concrete realisation of its p-adic motivic cohomology, thanks to results of Geisser-Levine and Bloch-Kato-Gabber which link them to algebraic K-theory. I will explain an analogous theory for formal schemes, as well as applications to algebraic cycles, such as a weak Lefschetz theorem for formal Chow groups.

(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.)

### 2015年12月09日(水)

18:00-19:00   数理科学研究科棟(駒場) 056号室
Ted Chinburg 氏 (University of Pennsylvania & IHES)
Chern classes in Iwasawa theory (English)
[ 講演概要 ]
Many of the main conjectures in Iwasawa theory can be phrased as saying that the first Chern class of an Iwasawa module is generated by a p-adic L-series. In this talk I will describe how higher Chern classes pertain to the higher codimension behavior of Iwasawa modules. I'll then describe a template for conjectures which would link such higher Chern classes to elements in the K-theory of Iwasawa algebras which are constructed from tuples of Katz p-adic L-series. I will finally describe an instance in which a result of this kind, for the second Chern class of an unramified Iwasawa module, can be proved over an imaginary quadratic field. This is joint work with F. Bleher, R. Greenberg, M. Kakde, G. Pappas, R. Sharifi and M. J. Taylor.

(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.)

### 2015年11月17日(火)

18:00-19:00   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ください
Dennis Gaitsgory 氏 (Harvard University & IHES)
The Tamagawa number formula over function fields. (English)
[ 講演概要 ]
Let G be a semi-simple and simply connected group and X an algebraic curve. We consider $Bun_G(X)$, the moduli space of G-bundles on X. In their celebrated paper, Atiyah and Bott gave a formula for the cohomology of $Bun_G$, namely $H^*(Bun_G)=Sym(H_*(X)\otimes V)$, where V is the space of generators for $H^*_G(pt)$. When we take our ground field to be a finite field, the Atiyah-Bott formula implies the Tamagawa number conjecture for the function field of X.

The caveat here is that the A-B proof uses the interpretation of $Bun_G$ as the space of connection forms modulo gauge transformations, and thus only works over complex numbers (but can be extend to any field of characteristic zero). In the talk we will outline an algebro-geometric proof that works over any ground field. As its main geometric ingredient, it uses the fact that the space of rational maps from X to G is homologically contractible. Because of the nature of the latter statement, the proof necessarily uses tools from higher category theory. So, it can be regarded as an example how the latter can be used to prove something concrete: a construction at the level of 2-categories leads to an equality of numbers.

(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.)

### 2015年10月27日(火)

18:00-19:00   数理科学研究科棟(駒場) 002号室

On the period conjecture of Gross-Deligne for fibrations (English)
[ 講演概要 ]
The period conjecture of Gross-Deligne asserts that the periods of algebraic varieties with complex multiplication are products of values of the gamma function at rational numbers. This is proved for CM elliptic curves by Lerch-Chowla-Selberg, and for abelian varieties by Shimura-Deligne-Anderson. However the question in the general case is still open. In this talk, we verify an alternating variant of the period conjecture for the cohomology of fibrations with relative multiplication. The proof relies on the Saito-Terasoma product formula for epsilon factors of integrable regular singular connections and the Riemann-Roch-Hirzebruch theorem. This is a joint work with Javier Fresan.

(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.)

### 2015年09月30日(水)

17:00-18:00   数理科学研究科棟(駒場) 056号室
Alan Lauder 氏 (University of Oxford)
Stark points and p-adic iterated integrals attached to modular forms of weight one (English)
[ 講演概要 ]
Given an elliptic curve over Q the only well-understood construction of global points is that of "Heegner points", which are defined over ring class fields of imaginary quadratic fields and are non-torsion only in rank one settings. I will present some new constructions and explicit formulae, in situations of rank one and two, of global points over ring class fields of real or imaginary quadratic fields, cyclotomic fields, and extensions of Q with Galois group A_4, S_4 or A_5. Our constructions and formulae are proven in certain cases - when they can be related to Heegner points - and conjectural, but supported by experimental evidence, otherwise. This is joint work with Henri Darmon and Victor Rotger.

### 2015年09月09日(水)

17:00-18:00   数理科学研究科棟(駒場) 056号室
Emmanuel Ullmo 氏 (IHES)
The hyperbolic Ax-Lindemann conjecture (English)
[ 講演概要 ]
The hyperbolic Ax Lindemann conjecture is a functional transcendental statement which describes the Zariski closure of "algebraic flows" on Shimura varieties. We will describe the proof of this conjecture and its consequences for the André-Oort conjecture. This is a joint work with Bruno Klingler and Andrei Yafaev.

### 2015年07月23日(木)

13:00-16:30   数理科学研究科棟(駒場) 056号室
Lasse Grimmelt 氏 (ゲッティンゲン大学/早稲田大学) 13:00-14:00
Representation of squares by cubic forms - Estimates for the appearing exponential sums (English)
Haoyu Hu 氏 (東京大学数理科学研究科) 14:15-15:15
Ramification and nearby cycles for $\ell$-adic sheaves on relative curves (English)
[ 講演概要 ]
I will present a new approach for a formula of Deligne and Kato that computes the dimension of the nearby cycle complex of an $\ell$-adic sheaf on a smooth relative curve over a strictly henselian trait such that $p$ is not one of its uniformizer. Deligne considered the case where the sheaf has no vertical ramification and Kato extended the formula to the general case. My approach is based on ramification theory of Abbes and Saito. It computes the nearby cycle complex in terms of the refined Swan conductor. In fact, I compare Abbes-Saito's refined Swan conductor with Kato's Swan conductor with differential values, which is the key ingredient in Kato's formula; the case of rank one sheaves is due to Abbes and Saito. My approach provides also a new independent proof of Deligne-Kato's formula.

Explicit computation of the number of dormant opers and duality (Japanese)

### 2015年06月17日(水)

17:00-18:00   数理科学研究科棟(駒場) 056号室

Hodge-Tate weights of p-adic Galois representations and Banach representations of GL_2(Q_p)
(Japanese)
[ 講演概要 ]
p進Galois表現のHodge-Tate重みを，p進Langlands対応により対応するGL_2(Q_p)のBanach表現から取り出すことが目標です．Banach表現の局所解析的ベクトルのなす空間へのLie環の作用からHodge-Tate重みが取り出せるというのが主結果で，Hodge-Tate表現の場合の局所代数的ベクトルについてのColmezの結果の解析的類似になっています．

### 2015年05月27日(水)

17:00-18:00   数理科学研究科棟(駒場) 056号室

On a good reduction criterion for polycurves with sections (Japanese)

### 2015年05月20日(水)

17:30-18:30   数理科学研究科棟(駒場) 056号室
Shou-Wu Zhang 氏 (Princeton University)
Colmez' conjecture in average (English)
[ 講演概要 ]
This is a report on a joint work with Xinyi Yuan on a conjectured formula of Colmez about the Faltings heights of CM abelian varieties. I will sketch a deduction of this formula in average of CM types from our early work on Gross-Zagier formula. When combined with a recent work of Tsimerman, this result implies the Andre-Oort conjecture for the moduli of abelian varieties.
Our method is different than a recently announced proof of a weaker form of the average formula by Andreatta, Howard, Goren, and Madapusi Pera: we use neither high dimensional Shimura varieties nor Borcherds' liftings.
(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.)