## 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

### 2020年06月17日(水)

17:30-18:30   オンライン開催
Christophe Breuil 氏 (CNRS, Université Paris-Sud)
On modular representations of GL_2(L) for unramified L (ENGLISH)
[ 講演概要 ]
Let p be a prime number and L a finite unramified extension of Q_p. We give a survey of past and new results on smooth admissible representations of GL_2(L) that appear in mod p cohomology. This is joint work with Florian Herzig, Yongquan Hu, Stefano Morra and Benjamin Schraen.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~t-saito/todai_IHES.html

### 2020年05月27日(水)

17:30-18:30   オンライン開催

Shintani generating class and the p-adic polylogarithm for totally real fields (ENGLISH)
[ 講演概要 ]
In this talk, we will give a new interpretation of Shintani's work concerning the generating function of nonpositive values of Hecke $L$-functions for totally real fields. In particular, we will construct a canonical class, which we call the Shintani generating class, in the cohomology of a certain quotient stack of an infinite direct sum of algebraic tori associated with a fixed totally real field. Using our observation that cohomology classes, not functions, play an important role in the higher dimensional case, we proceed to newly define the p-adic polylogarithm function in this case, and investigate its relation to the special value of p-adic Hecke $L$-functions. Some observations concerning the quotient stack will also be discussed. This is a joint work with Kei Hagihara, Kazuki Yamada, and Shuji Yamamoto.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~t-saito/todai_IHES.html

### 2020年05月13日(水)

17:30-18:30   オンライン開催
Yifeng Liu 氏 (Yale University)
On the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives (ENGLISH)
[ 講演概要 ]
In this talk, we will explain the final outcome on the Beilinson-Bloch-Kato conjecture for motives coming from certain automorphic representations of GL(n) x GL(n+1), of our recent project with Yichao Tian, Liang Xiao, Wei Zhang, and Xinwen Zhu. In particular, we show that the nonvanishing of the central L-value of the motive implies the vanishing of the corresponding Bloch-Kato Selmer group. We will also explain the main ideas and ingredients of the proof.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~t-saito/todai_IHES.html

### 2020年04月22日(水)

17:30-18:30   オンライン開催
Arthur-César Le Bras 氏 (CNRS & Université Paris 13)
Prismatic Dieudonné theory (ENGLISH)
[ 講演概要 ]
I would like to explain a classification result for p-divisible groups, which unifies many of the existing results in the literature. The main tool is the theory of prisms and prismatic cohomology recently developed by Bhatt and Scholze. This is joint work with Johannes Anschütz.

### 2019年12月04日(水)

17:00-18:00   数理科学研究科棟(駒場) 117号室

Characteristic epsilon cycles of l-adic sheaves on varieties (ENGLISH)
[ 講演概要 ]
For l-adic sheaves on varieties over finite fields, the constant terms of the functional equations of the L-functions, called global epsilon factors, are important arithmetic invariants. When the varieties are curves, Deligne and Laumon show that they admit product formulae in terms of local epsilon factors.
In this talk, I will explain that, attaching some coefficients to irreducible components of singular supports, we can define refinements of characteristic cycles. We will see that, after taking modulo roots of unity, they give product formulae of global epsilon factors for higher dimensional varieties.
I will also explain that these results can be generalized to arbitrary perfect fields of any characteristic.

### 2019年11月27日(水)

17:00-18:00   数理科学研究科棟(駒場) 117号室
Ahmed Abbes 氏 (CNRS & IHÉS)
The relative Hodge-Tate spectral sequence (ENGLISH)
[ 講演概要 ]
It is well known that the p-adic étale cohomology of a smooth and proper variety over a p-adic field admits a Hodge-Tate decomposition and that it is the abutment of a spectral sequence called Hodge-Tate; these two properties are incidentally equivalent. The Hodge-Tate decomposition was generalized in higher dimensions to Hodge-Tate local systems by Hyodo, and was studied by Faltings, Tsuji and others. But the generalization of the Hodge-Tate spectral sequence to a relative situation has not yet been considered (not even conjectured), with the exception of a special case of abelian schemes by Hyodo. This has now been done in a joint work with Michel Gros. The relative Hodge-Tate spectral sequence that we construct takes place in the Faltings topos, but its construction requires the introduction of a relative variant of this topos which is the main novelty of our work. The relative Hodge-Tate spectral sequence sheds new light on the fact that the relative p-adic étale cohomology is Hodge-Tate, but the two properties are not equivalent in general.

### 2019年11月20日(水)

18:00-19:00   数理科学研究科棟(駒場) 056号室
Vasudevan Srinivas 氏 (Tata Institute of Fundamental Research)
Algebraic versus topological entropy for surfaces over finite fields (ENGLISH)
[ 講演概要 ]
For an automorphism of an algebraic variety, we consider some properties of eigenvalues of the induced linear transformation on l-adic cohomology, motivated by some results from complex dynamics, related to the notion of entropy. This is a report on joint work with Hélène Esnault, and some subsequent work of K. Shuddhodan.

（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．今回は東京からの中継です．）

### 2019年10月16日(水)

17:30-18:30   数理科学研究科棟(駒場) 056号室
Liang Xiao 氏 (BICMR, Peking University)
On slopes of modular forms (ENGLISH)
[ 講演概要 ]
In this talk, I will survey some recent progress towards understanding the slopes of modular forms, with or without level structures. This has direct application to the conjecture of Breuil-Buzzard-Emerton on the slopes of Kisin's crystabelline deformation spaces. In particular, we obtain certain refined version of the spectral halo conjecture, where we may identify explicitly the slopes at the boundary when given a reducible non-split generic residual local Galois representation. This is a joint work in progress with Ruochuan Liu, Nha Truong, and Bin Zhao.

（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．）

### 2019年10月09日(水)

17:00-18:00   数理科学研究科棟(駒場) 056号室
Yuanqing Cai 氏 (京都大学)
Twisted doubling integrals for classical groups (ENGLISH)
[ 講演概要 ]
In the 1980s, Piatetski-Shapiro and Rallis discovered a family of Rankin-Selberg integrals for the classical groups that did not rely on Whittaker models. This is the so-called doubling method. It grew out of Rallis' work on the inner products of theta lifts -- the Rallis inner product formula.
In this talk, we present a family of Rankin-Selberg integrals (the twisted doubling method, in joint work with Friedberg, Ginzburg, and Kaplan) for the tensor product L-function of a pair of automorphic cuspidal representations, one of a classical group, the other of a general linear group. This can be viewed as a generalization of the doubling integrals of Piatetski-Shapiro and Rallis. Time permitting, we will discuss the twisted doubling integrals for Brylinski-Deligne covers of classical groups.

### 2019年07月03日(水)

17:00-18:00   数理科学研究科棟(駒場) 056号室

Explicit calculation of values of the regulator maps on a certain type of Kummer surfaces (Japanese)
[ 講演概要 ]

### 2019年06月12日(水)

17:00-18:00   数理科学研究科棟(駒場) 056号室

On extension of overconvergent log isocrystals on log smooth varieties (Japanese)
[ 講演概要 ]
Kを混標数の完備な非アルキメデス付値体とし，kをその剰余体とする．
Kedlayaおよび志甫の研究により，k上の滑らかな代数多様体Xとその上の単純正規交叉因子Zについて，X ¥setminus Z上の過収束アイソクリスタルのうちZの周りである種のモノドロミーを持つものは，XにZから定まる対数的構造を入れた対数的代数多様体上の収束対数的アイソクリスタルに延長できることが知られている．

### 2019年06月05日(水)

17:30-18:30   数理科学研究科棟(駒場) 056号室

Duality of Drinfeld modules and P-adic properties of Drinfeld modular forms (English)
[ 講演概要 ]
Let p be a rational prime, q>1 a p-power and P a non-constant irreducible polynomial in F_q[t]. The notion of Drinfeld modular form is an analogue over F_q(t) of that of elliptic modular form. Numerical computations suggest that Drinfeld modular forms enjoy some P-adic structures comparable to the elliptic analogue, while at present their P-adic properties are less well understood than the p-adic elliptic case. In 1990s, Taguchi established duality theories for Drinfeld modules and also for a certain class of finite flat group schemes called finite v-modules. Using the duality for the latter, we can define a function field analogue of the Hodge-Tate map. In this talk, I will explain how the Taguchi's theory and our Hodge-Tate map yield results on Drinfeld modular forms which are classical to elliptic modular forms e.g. P-adic congruences of Fourier coefficients imply p-adic congruences of weights.

### 2019年05月29日(水)

17:00-18:00   数理科学研究科棟(駒場) 056号室

On supersingular loci of Shimura varieties for quaternion unitary groups of degree 2 (Japanese)
[ 講演概要 ]
PEL型志村多様体のp進整数環上の整モデルは, Abel多様体と付加構造のモジュライ空間として定義される. その幾何的特殊ファイバーのうち, 超特異Abel多様体に対応する点からなる閉部分スキームを超特異部分という. 超特異部分の構造の明示的な記述は, arithmetic intersectionをはじめとする整数論への応用をもつことが知られている.

### 2019年05月08日(水)

17:00-18:00   数理科学研究科棟(駒場) 056号室

On the types for supercuspidal representations of inner forms of GL_n (Japanese)
[ 講演概要 ]
Aを非アルキメデス的局所体F上の中心的単純環とし，Gをその乗法群とする．
Gのsmooth表現を考察する際に有用な理論の一つとしてtypeの理論が存在する．
type (J, ¥lambda) とはGのコンパクト部分群Jと J の既約部分表現 ¥lambda の組であって，Gの既約表現をある意味で分類することのできるものである．
S¥'echerre-Stevenにより，Gのtypeの族としてsimple typeという概念が構成されている．

### 2019年04月30日(火)

17:00-18:00   数理科学研究科棟(駒場) 122号室
Jean-Francois Dat 氏 (Sorbonne University)
Moduli space of l-adic Langlands parameters and the stable Bernstein center (English)
[ 講演概要 ]
Motivated by the description of the integral l-adic cohomology of certain Shimura varieties in middle degree, Emerton and Helm have conjectured the existence of a certain local Langlands correspondence for l-adic families of n-dimensional Galois representations. The proof of this conjecture by Helm and Moss relies on a beautiful isomorphism between the ring of functions of the moduli space of l-adic representations and the integral Bernstein center of GL_n(F). We will present a work in progress with Helm, Korinczuk and Moss towards a generalization of this result for arbitrary (tamely ramified) reductive groups.

### 2019年04月24日(水)

17:30-18:30   数理科学研究科棟(駒場) 056号室
Joseph Ayoub 氏 (University of Zurich)
P^1-localisation and a possible definition of arithmetic Kodaira-Spencer classes (English)
[ 講演概要 ]
A^1-localisation is a universal construction which produces "cohomology theories" for which the affine line A^1 is contractible. It plays a central role in the theory of motives à la Morel-Voevodsky. In this talk, I'll discuss the analogous construction where the affine line is replaced by the projective line P^1. This is the P^1-localisation which is arguably an unnatural construction since it produces "cohomology theories" for which the projective line P^1 is contractible. Nevertheless, I'll explain a few positive results and some computations around this construction which naturally lead to a definition of Kodaira-Spencer classes of arithmetic nature. (Unfortunately, it is yet unclear if these classes are really interesting and nontrivial.)

### 2019年04月17日(水)

17:00-18:00   数理科学研究科棟(駒場) 122号室

On the Shafarevich conjecture for minimal surfaces of Kodaira dimension 0 (Japanese)
[ 講演概要 ]
Fを代数体、Sを有限素点の有限集合とする。
Faltings-Zarhinは、固定した正整数gに対して、
Sの外で良還元を持つようなF上のg次元Abel多様体のF上の同型類の有限性を証明した。
(Abel多様体のShafarevich予想)

また、超楕円曲面の還元に関連する話題として、

### 2019年04月10日(水)

17:30-18:30   数理科学研究科棟(駒場) 056号室
Zongbin Chen 氏 (丘成桐数学科学中心, 清華大学)
The geometry of the affine Springer fibers and Arthur's weighted orbital integrals (English)
[ 講演概要 ]
The affine Springer fibers are geometric objects conceived for the study of orbital integrals. They have complicated geometric structures. We will explain our work on the geometry of affine Springer fibers, with emphasize on the construction of a fundamental domain, and show how the study of the affine Springer fibers can be reduced to that of its fundamental domain. As an application, we will explain how to calculate Arthur's weighted orbital integrals via counting points on the fundamental domain.

（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．今回は東京からの中継です．）

### 2019年01月16日(水)

18:00-19:00   数理科学研究科棟(駒場) 056号室
Lei Fu 氏 (Yau Mathematical Sciences Center, Tsinghua University)
[ 講演概要 ]
Using Dwork's trace formula, we express the exponential sum associated to a Laurent polynomial as the trace of a chain map on a twisted de Rham complex for the torus over the p-adic field. Treating the coefficients of the polynomial as parameters, we obtain the p-adic Gelfand-Kapranov-Zelevinsky (GKZ) system, which is a complex of D^\dagger-modules with Frobenius structure.

（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．）

### 2019年01月09日(水)

17:00-18:00   数理科学研究科棟(駒場) 056号室
Laurent Berger 氏 (ENS de Lyon)
Formal groups and p-adic dynamical systems (ENGLISH)
[ 講演概要 ]
A formal group gives rise to a p-adic dynamical system. I will discuss some results about formal groups that can be proved using this point of view. I will also discuss the theory of p-adic dynamical systems and some open questions.

### 2018年12月19日(水)

17:30-18:30   数理科学研究科棟(駒場) 056号室
Jean-Stefan Koskivirta 氏 (東京大学数理科学研究科)
Cohomology vanishing for automorphic vector bundles (ENGLISH)
[ 講演概要 ]
A Shimura variety carries naturally a family of vector bundles parametrized by the characters of a maximal torus in the attached group. We want to determine which of these vector bundles are ample, and also show cohomology vanishing results. For this we use generalized Hasse invariants on the stack of G-zips of Moonen-Pink-Wedhorn-Ziegler. It is a group-theoretical counterpart of the Shimura variety and carries a similar family of vector bundles. This is joint work with Y.Brunebarbe, W.Goldring and B.Stroh.

### 2018年12月12日(水)

18:00-19:00   数理科学研究科棟(駒場) 056号室
Gaëtan Chenevier 氏 (CNRS, Université Paris-Sud)
A higher weight (and automorphic) generalization of the Hermite-Minkowski theorem (ENGLISH)
[ 講演概要 ]
I will show that for any integer N, there are only finitely many cuspidal algebraic automorphic representations of GL_m over Q whose Artin conductor is N and whose "weights" are in the interval {0,...,23} (with m varying). Via the conjectural yoga between geometric Galois representations (or motives) and algebraic automorphic forms, this statement may be viewed as a generalization of the classical Hermite-Minkowski theorem in algebraic number theory. I will also discuss variants of these results when the base field Q is replaced by an arbitrary number field.

（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．）

### 2018年11月21日(水)

17:00-18:00   数理科学研究科棟(駒場) 056号室
Yves André 氏 (Université Pierre et Marie Curie)
Poncelet games, confinement of algebraic integers, and hyperbolic Ax-Schanuel (ENGLISH)
[ 講演概要 ]
We shall theorize and exemplify the problem of torsion values of sections of abelian schemes. This « unlikely intersection problem », which arises in various diophantine and algebro-geometric contexts, can be reformulated in a non-trivial way in terms of Kodaira-Spencer maps. A key tool toward its general solution is then provided by recent theorems of Ax-Schanuel type (joint work with P. Corvaja, U. Zannier, and partly Z. Gao).

### 2018年11月14日(水)

18:00-19:00   数理科学研究科棟(駒場) 056号室

A motivic construction of ramification filtrations (ENGLISH)
[ 講演概要 ]
We give a new interpretation of Artin conductors of characters in the framework of theory of motives with modulus. It gives a unified way to understand Artin conductors of characters and irregularities of line bundle with integrable connections as well as overconvergent F-isocrystals of rank 1. It also gives rise to new conductors, for example, for G-torsors with G a finite flat group scheme, which specializes to the classical Artin conductor in case G = Z/nZ. We also give a motivic proof of a theorem of Kato and Matsuda on the determination of Artin conductors along divisors on smooth schemes by its restrictions to curves. Its proof is based on a motivic version of a theorem of Gabber-Katz. This is a joint work with Kay Rülling.

### 2018年10月10日(水)

18:00-19:00   数理科学研究科棟(駒場) 056号室
Yichao Tian 氏 (Université de Strasbourg)
Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives (ENGLISH)
[ 講演概要 ]
In my talk, I will report on my ongoing collaborating project together with Yifeng Liu, Liang Xiao, Wei Zhang, and Xinwen Zhu, which concerns the rank 0 case of the Beilinson-Bloch-Kato conjecture on the relation between L-functions and Selmer groups for certain Rankin--Selberg motives for GL(n) x GL(n+1). I will state the main results with some examples coming from elliptic curves, sketch the strategy of the proof, and then focus on the key geometric ingredients, namely the semi-stable reduction of unitary Shimura varieties of type U(1,n) at non-quasi-split places.

（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．）