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過去の記録 ～05/21｜次回の予定｜今後の予定 05/22～
| 開催情報 | 水曜日 17:00～18:00 数理科学研究科棟(駒場) 117号室 |
|---|---|
| 担当者 | 今井 直毅，ケリー シェーン |
過去の記録
2025年05月21日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Toni Annala 氏 (University of Chicago)
A¹-Colocalization and Logarithmic Cohomology Theories
https://tannala.com/
Toni Annala 氏 (University of Chicago)
A¹-Colocalization and Logarithmic Cohomology Theories
[ 講演概要 ]
In recent joint work with Hoyois and Iwasa, we discovered that non-A¹-invariant motivic homotopy theory offers a new lens for understanding logarithmic cohomology theories. Central to this perspective is A¹-colocalization, which produces a cohomology theory whose value on a smooth scheme U agrees with the "logarithmic cohomology" of a good compactification (X,D). In many examples, including de Rham and crystalline cohomology, the quotation marks can be dropped, as A¹-colocalization recovers the classical logarithmic cohomology groups. I will explain this connection and, time permitting, sketch a proof of the duality theorem underlying this phenomenon, which states that smooth projective schemes have a dualizable motive.
[ 参考URL ]In recent joint work with Hoyois and Iwasa, we discovered that non-A¹-invariant motivic homotopy theory offers a new lens for understanding logarithmic cohomology theories. Central to this perspective is A¹-colocalization, which produces a cohomology theory whose value on a smooth scheme U agrees with the "logarithmic cohomology" of a good compactification (X,D). In many examples, including de Rham and crystalline cohomology, the quotation marks can be dropped, as A¹-colocalization recovers the classical logarithmic cohomology groups. I will explain this connection and, time permitting, sketch a proof of the duality theorem underlying this phenomenon, which states that smooth projective schemes have a dualizable motive.
https://tannala.com/
2025年05月14日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Eamon Quinlan 氏 (University of Utah)
Introduction to the Bernstein-Sato polynomial in positive characteristic
https://eamonqg.github.io/
Eamon Quinlan 氏 (University of Utah)
Introduction to the Bernstein-Sato polynomial in positive characteristic
[ 講演概要 ]
The Bernstein-Sato polynomial of a holomorphic function is an invariant that originated in complex analysis, and with now strong applications to birational geometry and singularity theory over the complex numbers. For example, it detects the log-canonical threshold as well as the eigenvalues of the monodromy action on nearby cycles. In this talk I will define a characteristic-p analogue of this invariant, I will survey some of its basic properties, and I will illustrate how its behavior reflects arithmetic phenomena. This will serve as an introduction to the talk by Hiroki Kato.
[ 参考URL ]The Bernstein-Sato polynomial of a holomorphic function is an invariant that originated in complex analysis, and with now strong applications to birational geometry and singularity theory over the complex numbers. For example, it detects the log-canonical threshold as well as the eigenvalues of the monodromy action on nearby cycles. In this talk I will define a characteristic-p analogue of this invariant, I will survey some of its basic properties, and I will illustrate how its behavior reflects arithmetic phenomena. This will serve as an introduction to the talk by Hiroki Kato.
https://eamonqg.github.io/
2025年05月14日(水)
18:10-19:10 数理科学研究科棟(駒場) 117号室
加藤大輝 氏 (IHES)
Bernstein--Sato theory in positive characteristic and unit root nearby cycles.
加藤大輝 氏 (IHES)
Bernstein--Sato theory in positive characteristic and unit root nearby cycles.
[ 講演概要 ]
I will talk about how to formulate (and outline an idea of a proof of) a positive characteristic analogue of the theorem of Kashiwara and Malgrange about the relationship, in characteristic zero, between the Bernstein-Sato polynomial and the eigenvalues of the monodromy action on nearby cycles. It will/is expected to give a cohomological explanation for some of the arithmetic phenomena that will be presented in the talk by Eamon Quinlan. This is a joint work in progress with him and Daichi Takeuchi.
I will talk about how to formulate (and outline an idea of a proof of) a positive characteristic analogue of the theorem of Kashiwara and Malgrange about the relationship, in characteristic zero, between the Bernstein-Sato polynomial and the eigenvalues of the monodromy action on nearby cycles. It will/is expected to give a cohomological explanation for some of the arithmetic phenomena that will be presented in the talk by Eamon Quinlan. This is a joint work in progress with him and Daichi Takeuchi.
2025年05月07日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Eric Chen 氏 (École Polytechnique Fédérale de Lausanne (EPFL))
Hanany–Witten transition and Rankin–Selberg periods
https://sites.google.com/view/eric-yen-yo-chen-math/homepage
Eric Chen 氏 (École Polytechnique Fédérale de Lausanne (EPFL))
Hanany–Witten transition and Rankin–Selberg periods
[ 講演概要 ]
Given a manifold decorated with defects separating the bulk into regions with varying gauge theories, it is often convenient to first simplify the topological picture into model configurations. One of the simplest examples of such operations are the transition moves introduced by Hanany and Witten, which describes the crossing of an NS5 and a D5 brane. In this talk, we will describe how these ideas lead to identities between Rankin—Selberg type integrals over moduli spaces of bundles, and consequences of S-duality, or relative Langlands duality, in this setup.
[ 参考URL ]Given a manifold decorated with defects separating the bulk into regions with varying gauge theories, it is often convenient to first simplify the topological picture into model configurations. One of the simplest examples of such operations are the transition moves introduced by Hanany and Witten, which describes the crossing of an NS5 and a D5 brane. In this talk, we will describe how these ideas lead to identities between Rankin—Selberg type integrals over moduli spaces of bundles, and consequences of S-duality, or relative Langlands duality, in this setup.
https://sites.google.com/view/eric-yen-yo-chen-math/homepage
2025年04月23日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Dat Pham 氏 (C.N.R.S., IMJ-PRG, Sorbonne Université)
Prismatic F-crystals and "Lubin--Tate" crystalline Galois representations.
https://webusers.imj-prg.fr/~dat.pham/
Dat Pham 氏 (C.N.R.S., IMJ-PRG, Sorbonne Université)
Prismatic F-crystals and "Lubin--Tate" crystalline Galois representations.
[ 講演概要 ]
An important question in integral p-adic Hodge theory is the study of lattices in crystalline Galois representations. There have been various classifications of such objects, such as Fontaine--Lafaille’s theory, Breuil’s theory of strongly divisible lattices, and Kisin’s theory of Breuil--Kisin modules. Using their prismatic theory, Bhatt--Scholze give a site-theoretic description of such lattices, which has the nice feature that it can specialize to many of the previous classifications by "evaluating" suitably. In this talk, we will recall their result and explain an extension to the Lubin--Tate context.
[ 参考URL ]An important question in integral p-adic Hodge theory is the study of lattices in crystalline Galois representations. There have been various classifications of such objects, such as Fontaine--Lafaille’s theory, Breuil’s theory of strongly divisible lattices, and Kisin’s theory of Breuil--Kisin modules. Using their prismatic theory, Bhatt--Scholze give a site-theoretic description of such lattices, which has the nice feature that it can specialize to many of the previous classifications by "evaluating" suitably. In this talk, we will recall their result and explain an extension to the Lubin--Tate context.
https://webusers.imj-prg.fr/~dat.pham/
2024年12月11日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
大江亮輔 氏 (東京大学大学院数理科学研究科)
The characteristic cycle of an l-adic sheaf on a smooth variety (Japanese)
大江亮輔 氏 (東京大学大学院数理科学研究科)
The characteristic cycle of an l-adic sheaf on a smooth variety (Japanese)
[ 講演概要 ]
The characteristic cycle of an l-adic sheaf on a smooth variety over a perfect field is defined by Saito as a cycle on the cotangent bundle and the intersection with the zero section computes the Euler number. On the other hand, the characteristic cycle of an l-adic sheaf on a regular scheme in mixed characteristic is not yet defined. In this talk, I define the F-characteristic cycle of a rank one sheaf on an arithmetic surface whose intersection with the zero section computes the Swan conductor of the cohomology of the generic fiber. The definition is based on the computation of the characteristic cycle in equal characteristic by Yatagawa. I explain the rationality and the integrality of the characteristic form of an abelian character, which are necessary for the definition of the F-characteristic cycle.
The characteristic cycle of an l-adic sheaf on a smooth variety over a perfect field is defined by Saito as a cycle on the cotangent bundle and the intersection with the zero section computes the Euler number. On the other hand, the characteristic cycle of an l-adic sheaf on a regular scheme in mixed characteristic is not yet defined. In this talk, I define the F-characteristic cycle of a rank one sheaf on an arithmetic surface whose intersection with the zero section computes the Swan conductor of the cohomology of the generic fiber. The definition is based on the computation of the characteristic cycle in equal characteristic by Yatagawa. I explain the rationality and the integrality of the characteristic form of an abelian character, which are necessary for the definition of the F-characteristic cycle.
2024年12月04日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Kieu Hieu Nguyen 氏 (University of Versailles Saint-Quentin)
On categorical local Langlands for GLn (English)
Kieu Hieu Nguyen 氏 (University of Versailles Saint-Quentin)
On categorical local Langlands for GLn (English)
[ 講演概要 ]
Recently, Fargues-Scholze and many other people realized that there should be a categorical version which encodes great information of the local Langlands correspondence. In this talk, I will describe the objects appearing in their conjectures and explain some relations with the local Langlands correspondences for GLn.
Recently, Fargues-Scholze and many other people realized that there should be a categorical version which encodes great information of the local Langlands correspondence. In this talk, I will describe the objects appearing in their conjectures and explain some relations with the local Langlands correspondences for GLn.
2024年11月27日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
桝澤海斗 氏 (東京大学大学院数理科学研究科)
On the correspondence of simple supercuspidal representations of $\mathrm{GSp}_{2n}$ and its inner form (Japanese)
桝澤海斗 氏 (東京大学大学院数理科学研究科)
On the correspondence of simple supercuspidal representations of $\mathrm{GSp}_{2n}$ and its inner form (Japanese)
[ 講演概要 ]
Let $F$ be a nonarchimedean local field. The local Jacquet-Langlands correspondence is the one-to-one correspondence of essential square integrable representations of $\mathrm{GL}_n(F)$ and its inner form. It is known that this correspondence satisfies the character relation and preserves the simple supercuspidality. We assume the correspondence of simple supercuspidal representations of $\mathrm{GSp}_{2n}(F)$ and irreducible admissible representations of its inner form which satisfies the character relation. This is expected to exist by a standard argument using the theory of stable trace formula. In this talk, we show the simple supercuspidality is preserved under this correspondence. In addition, we can parametrize simple supercuspidal representations and describe the correspondence explicitly.
Let $F$ be a nonarchimedean local field. The local Jacquet-Langlands correspondence is the one-to-one correspondence of essential square integrable representations of $\mathrm{GL}_n(F)$ and its inner form. It is known that this correspondence satisfies the character relation and preserves the simple supercuspidality. We assume the correspondence of simple supercuspidal representations of $\mathrm{GSp}_{2n}(F)$ and irreducible admissible representations of its inner form which satisfies the character relation. This is expected to exist by a standard argument using the theory of stable trace formula. In this talk, we show the simple supercuspidality is preserved under this correspondence. In addition, we can parametrize simple supercuspidal representations and describe the correspondence explicitly.
2024年11月06日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Piotr Pstrągowski 氏 (京都大学)
The even filtration and prismatic cohomology (English)
Piotr Pstrągowski 氏 (京都大学)
The even filtration and prismatic cohomology (English)
[ 講演概要 ]
The even filtration, introduced by Hahn-Raksit-Wilson, is a canonical filtration attached to a commutative ring spectrum which measures its failure to be even. Despite its simple definition, the even filtration recovers many arithmetically important constructions, such as the Adams-Novikov filtration of the sphere or the Bhatt-Morrow-Scholze filtration on topological Hochschild homology, showing that they are all invariants of the commutative ring spectrum alone. I will describe a linear variant of the even filtration which is naturally defined on associative rings as well as joint work with Raksit on the resulting extension of prismatic cohomology to the context of E_2-rings.
The even filtration, introduced by Hahn-Raksit-Wilson, is a canonical filtration attached to a commutative ring spectrum which measures its failure to be even. Despite its simple definition, the even filtration recovers many arithmetically important constructions, such as the Adams-Novikov filtration of the sphere or the Bhatt-Morrow-Scholze filtration on topological Hochschild homology, showing that they are all invariants of the commutative ring spectrum alone. I will describe a linear variant of the even filtration which is naturally defined on associative rings as well as joint work with Raksit on the resulting extension of prismatic cohomology to the context of E_2-rings.
2024年10月30日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
佐藤匡弥 氏 (東京大学大学院数理科学研究科)
Representability of Hochschild homology in the category of motives with modulus (日本語)
佐藤匡弥 氏 (東京大学大学院数理科学研究科)
Representability of Hochschild homology in the category of motives with modulus (日本語)
[ 講演概要 ]
There is a map from algebraic K-theory to Hochschild homology, called a trace map. This map developed the study of algebraic K-theory. Algebraic K-theory is A^1-invariant on the category of smooth schemes over a field, so the Voevodsky’s motivic homotopy theory is a nice way to study algebraic K-theory. However, Hochschild homology is not A^1-invariant, so Voevodsky’s theory doesn’t capture it. In this talk, we will extend Hochschild homology of schemes to modulus pairs, and it is representable in the category of motives with modulus defined by Kahn-Miyazaki-Saito-Yamazaki.
There is a map from algebraic K-theory to Hochschild homology, called a trace map. This map developed the study of algebraic K-theory. Algebraic K-theory is A^1-invariant on the category of smooth schemes over a field, so the Voevodsky’s motivic homotopy theory is a nice way to study algebraic K-theory. However, Hochschild homology is not A^1-invariant, so Voevodsky’s theory doesn’t capture it. In this talk, we will extend Hochschild homology of schemes to modulus pairs, and it is representable in the category of motives with modulus defined by Kahn-Miyazaki-Saito-Yamazaki.
2024年10月16日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Pierre Colmez 氏 (Sorbonne University)
On the factorisation of Beilinson-Kato system (English)
Pierre Colmez 氏 (Sorbonne University)
On the factorisation of Beilinson-Kato system (English)
[ 講演概要 ]
I will explain how one can factor Beilinson-Kato system as a product of two modular symbols, an algebraic incarnation of Rankin's method. This is joint work with Shanwen Wang.
I will explain how one can factor Beilinson-Kato system as a product of two modular symbols, an algebraic incarnation of Rankin's method. This is joint work with Shanwen Wang.
2024年10月02日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Hui Gao 氏 (Southern University of Science and Technology)
Filtered integral Sen theory (English)
Hui Gao 氏 (Southern University of Science and Technology)
Filtered integral Sen theory (English)
[ 講演概要 ]
Using the Breuil--Kisin module attached to an integral crystalline representation, one can define an integral Hodge filtration whose behavior is closely related to arithmetic and geometry of the representation. In this talk, we discuss vanishing and torsion bound on graded pieces of this filtration, using a filtered integral Sen theory as key tool. This is joint work with Tong Liu.
Using the Breuil--Kisin module attached to an integral crystalline representation, one can define an integral Hodge filtration whose behavior is closely related to arithmetic and geometry of the representation. In this talk, we discuss vanishing and torsion bound on graded pieces of this filtration, using a filtered integral Sen theory as key tool. This is joint work with Tong Liu.
2024年09月18日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Ishan Levy 氏 (University of Copenhagen)
Telescopic stable homotopy theory (English)
Ishan Levy 氏 (University of Copenhagen)
Telescopic stable homotopy theory (English)
[ 講演概要 ]
Chromatic homotopy theory attempts to study the stable homotopy category by breaking it into v_n-periodic layers corresponding to height n formal groups. There are two natural ways to do this, via either the K(n)-localizations which are computationally accessible, or via the T(n)-localizations, which detect the v_n-periodic parts of the stable homotopy groups of spheres. Ravenel's telescope conjecture asks that these two localizations agree. For n at least 2 and all primes, I will discuss counterexamples to Ravenel’s telescope conjecture. Our counterexamples come from using trace methods to compute the T(n) and K(n)-localizations of the algebraic K-theory of a family of ring spectra, which in the case n=2 are certain finite Galois extensions of the K(1)-local sphere. I will then explain that this can be used to obtain an infinite family of elements in the v_n-periodic stable homotopy groups of spheres, giving the best known lower bound on the asymptotic average ranks of the stable stems. Finally, I will explain that the Galois group of the T(n)-local category agrees with that of the K(n)-local category, and how the failure of the telescope conjecture comes entirely from the failure of Galois hyperdescent. This talk comes from projects that are joint with Burklund, Carmeli, Clausen, Hahn, Schlank, and Yanovski.
Chromatic homotopy theory attempts to study the stable homotopy category by breaking it into v_n-periodic layers corresponding to height n formal groups. There are two natural ways to do this, via either the K(n)-localizations which are computationally accessible, or via the T(n)-localizations, which detect the v_n-periodic parts of the stable homotopy groups of spheres. Ravenel's telescope conjecture asks that these two localizations agree. For n at least 2 and all primes, I will discuss counterexamples to Ravenel’s telescope conjecture. Our counterexamples come from using trace methods to compute the T(n) and K(n)-localizations of the algebraic K-theory of a family of ring spectra, which in the case n=2 are certain finite Galois extensions of the K(1)-local sphere. I will then explain that this can be used to obtain an infinite family of elements in the v_n-periodic stable homotopy groups of spheres, giving the best known lower bound on the asymptotic average ranks of the stable stems. Finally, I will explain that the Galois group of the T(n)-local category agrees with that of the K(n)-local category, and how the failure of the telescope conjecture comes entirely from the failure of Galois hyperdescent. This talk comes from projects that are joint with Burklund, Carmeli, Clausen, Hahn, Schlank, and Yanovski.
2024年07月10日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Chieh-Yu Chang 氏 (National Tsing Hua University)
On special v-adic gamma values after Gross-Koblitz-Thakur (英語)
Chieh-Yu Chang 氏 (National Tsing Hua University)
On special v-adic gamma values after Gross-Koblitz-Thakur (英語)
[ 講演概要 ]
In this talk, we will introduce special v-adic arithmetic gamma values in positive characteristic, which play the function field analogue of the special values of Morita’s p-adic gamma function. In the function field case, Thakur established a formula à la Gross-Koblitz, and hence obtained algebraicity of certain special v-adic arithmetic gamma values. In a joint work with Fu-Tsun Wei and Jing Yu, we prove that all algebraic relations among these special v-adic gamma values are coming from the three types of functional equations that the v-adic arithmetic gamma function satisfies, and Thakur’s analogue of Gross-Koblitz’s formula.
In this talk, we will introduce special v-adic arithmetic gamma values in positive characteristic, which play the function field analogue of the special values of Morita’s p-adic gamma function. In the function field case, Thakur established a formula à la Gross-Koblitz, and hence obtained algebraicity of certain special v-adic arithmetic gamma values. In a joint work with Fu-Tsun Wei and Jing Yu, we prove that all algebraic relations among these special v-adic gamma values are coming from the three types of functional equations that the v-adic arithmetic gamma function satisfies, and Thakur’s analogue of Gross-Koblitz’s formula.
2024年06月19日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Abhinandan 氏 (University of Tokyo)
Prismatic $F$-crystals and Wach modules (English)
Abhinandan 氏 (University of Tokyo)
Prismatic $F$-crystals and Wach modules (English)
[ 講演概要 ]
For an absolutely unramified extension $K/\mathbb{Q}_p$ with perfect residue field, by the works of Fontaine, Colmez, Wach and Berger, it is well known that the category of Wach modules over a certain integral period ring is equivalent to the category of lattices inside crystalline representations of $G_K$ (the absolute Galois group of $K$). Moreover, by the recent works of Bhatt and Scholze, we also know that lattices inside crystalline representations of $G_K$ are equivalent to the category of prismatic $F$-crystals on the absolute prismatic site of $O_K$, the ring of integers of $K$. The goal of this talk is to present a direct construction of the categorical equivalence between Wach modules and prismatic $F$-crystals over the absolute prismatic site of $O_K$. If time permits, we will also mention a generalisation of these results to the case of a "small" base ring.
For an absolutely unramified extension $K/\mathbb{Q}_p$ with perfect residue field, by the works of Fontaine, Colmez, Wach and Berger, it is well known that the category of Wach modules over a certain integral period ring is equivalent to the category of lattices inside crystalline representations of $G_K$ (the absolute Galois group of $K$). Moreover, by the recent works of Bhatt and Scholze, we also know that lattices inside crystalline representations of $G_K$ are equivalent to the category of prismatic $F$-crystals on the absolute prismatic site of $O_K$, the ring of integers of $K$. The goal of this talk is to present a direct construction of the categorical equivalence between Wach modules and prismatic $F$-crystals over the absolute prismatic site of $O_K$. If time permits, we will also mention a generalisation of these results to the case of a "small" base ring.
2024年05月22日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
渡部匠 氏 (東京大学大学院数理科学研究科)
On the (φ,Γ)-modules corresponding to crystalline representations and semi-stable representations
渡部匠 氏 (東京大学大学院数理科学研究科)
On the (φ,Γ)-modules corresponding to crystalline representations and semi-stable representations
[ 講演概要 ]
From the 1980s to the 1990s, J.-M. Fontaine constructed an equivalence of categories between the category of (φ, Γ)-modules and the category of p-adic Galois representations. After recalling it, I will present my result on the (φ, Γ)-modules corresponding to crystalline representations and semi-stable representations. As for the crystalline case, this can be seen, in a sense, as a generalization of Wach module in the ramified case. If time permits, I will explain my ongoing research on the (φ, Γ)-modules corresponding to de Rham representations.
From the 1980s to the 1990s, J.-M. Fontaine constructed an equivalence of categories between the category of (φ, Γ)-modules and the category of p-adic Galois representations. After recalling it, I will present my result on the (φ, Γ)-modules corresponding to crystalline representations and semi-stable representations. As for the crystalline case, this can be seen, in a sense, as a generalization of Wach module in the ramified case. If time permits, I will explain my ongoing research on the (φ, Γ)-modules corresponding to de Rham representations.
2024年05月15日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
高谷悠太 氏 (東京大学大学院数理科学研究科)
Equidimensionality of affine Deligne-Lusztig varieties in mixed characteristic (日本語)
高谷悠太 氏 (東京大学大学院数理科学研究科)
Equidimensionality of affine Deligne-Lusztig varieties in mixed characteristic (日本語)
[ 講演概要 ]
Shimura varieties are of central interest in arithmetic geometry and affine Deligne-Lusztig varieties are closely related to their special fibers. These varieties are group-theoretical objects and can be defined even for non-miniscule local Shimura data. In this talk, I will explain the proof of the equidimensionality of affine Deligne-Lusztig varieties in mixed characteristic.
The main ingredient is a local foliation of affine Deligne-Lusztig varieties in mixed characteristic. In equal characteristic, this local structure was previously introduced by Hartl and Viehmann.
Shimura varieties are of central interest in arithmetic geometry and affine Deligne-Lusztig varieties are closely related to their special fibers. These varieties are group-theoretical objects and can be defined even for non-miniscule local Shimura data. In this talk, I will explain the proof of the equidimensionality of affine Deligne-Lusztig varieties in mixed characteristic.
The main ingredient is a local foliation of affine Deligne-Lusztig varieties in mixed characteristic. In equal characteristic, this local structure was previously introduced by Hartl and Viehmann.
2024年05月08日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Xinyao Zhang 氏 (東京大学大学院数理科学研究科)
The pro-modularity in the residually reducible case (English)
Xinyao Zhang 氏 (東京大学大学院数理科学研究科)
The pro-modularity in the residually reducible case (English)
[ 講演概要 ]
For a continuous odd two dimensional Galois representation over a finite field of characteristic p, it is conjectured that its universal deformation ring is isomorphic to some p-adic big Hecke algebra, called the big R=T theorem. Recently, Deo explored the residually reducible case and proved a big R=T theorem for Q under the assumption of the cyclicity of some cohomology group. However, his method is unavailable for totally real fields since the assumption does not hold any longer. In this talk, we follow the strategy of the work from Skinner-Wiles and Pan on the Fontaine-Mazur conjecture and give a pro-modularity result for some totally real fields, which is an analogue to the big R=T theorem.
For a continuous odd two dimensional Galois representation over a finite field of characteristic p, it is conjectured that its universal deformation ring is isomorphic to some p-adic big Hecke algebra, called the big R=T theorem. Recently, Deo explored the residually reducible case and proved a big R=T theorem for Q under the assumption of the cyclicity of some cohomology group. However, his method is unavailable for totally real fields since the assumption does not hold any longer. In this talk, we follow the strategy of the work from Skinner-Wiles and Pan on the Fontaine-Mazur conjecture and give a pro-modularity result for some totally real fields, which is an analogue to the big R=T theorem.
2024年05月01日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
松本晃二郎 氏 (東京大学大学院数理科学研究科)
On the potential automorphy and the local-global compatibility for the monodromy operators at p ≠ l over CM fields. (日本語)
松本晃二郎 氏 (東京大学大学院数理科学研究科)
On the potential automorphy and the local-global compatibility for the monodromy operators at p ≠ l over CM fields. (日本語)
[ 講演概要 ]
Let F be a totally real field or CM field, n be a positive integer, l be a prime, π be a cohomological cuspidal automorphic representation of GLn over F and v be a non-l-adic finite place of F. In 2014, Harris-Lan-Taylor-Thorne constructed the l-adic Galois representation corresponding to π. (Scholze also constructed this by another method.) The compatibility of this construction and the local Langlands correspondence at v was proved up to semisimplification by Ila Varma(2014), but the compatibility for the monodromy operators was known only in conjugate self-dual cases and some special 2-dimensional cases. In this talk, we will prove the local-global compatibility in some self-dual cases and sufficiently regular weight cases by using some new potential automorphy theorems. Moreover, if we have time, we will also prove the Ramanujan conjecture for the cohomological cuspidal automorphic representations of GL2 over F, which was proved in parallel weight cases by Boxer-Calegari-Gee-Newton-Thorne (2023).
Let F be a totally real field or CM field, n be a positive integer, l be a prime, π be a cohomological cuspidal automorphic representation of GLn over F and v be a non-l-adic finite place of F. In 2014, Harris-Lan-Taylor-Thorne constructed the l-adic Galois representation corresponding to π. (Scholze also constructed this by another method.) The compatibility of this construction and the local Langlands correspondence at v was proved up to semisimplification by Ila Varma(2014), but the compatibility for the monodromy operators was known only in conjugate self-dual cases and some special 2-dimensional cases. In this talk, we will prove the local-global compatibility in some self-dual cases and sufficiently regular weight cases by using some new potential automorphy theorems. Moreover, if we have time, we will also prove the Ramanujan conjecture for the cohomological cuspidal automorphic representations of GL2 over F, which was proved in parallel weight cases by Boxer-Calegari-Gee-Newton-Thorne (2023).
2024年04月17日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Ahmed Abbes 氏 (IHES、東大数理（日本学術振興会 外国人招へい研究者）)
Functoriality of the p-adic Simpson correspondence by proper push forward (English)
Ahmed Abbes 氏 (IHES、東大数理（日本学術振興会 外国人招へい研究者）)
Functoriality of the p-adic Simpson correspondence by proper push forward (English)
[ 講演概要 ]
Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence whose construction has been taken up by various authors, according to several approaches. After recalling the one initiated by myself with Michel Gros, I will present our initial result on the functoriality of the p-adic Simpson correspondence by proper push forward, leading to a generalization of the relative Hodge-Tate spectral sequence. If time permits, I will give a brief overview of an ongoing project with Michel Gros and Takeshi Tsuji, aimed at establishing a more robust framework for achieving broader functoriality results of the p-adic Simpson correspondence, by both proper push forward and pullback.
Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence whose construction has been taken up by various authors, according to several approaches. After recalling the one initiated by myself with Michel Gros, I will present our initial result on the functoriality of the p-adic Simpson correspondence by proper push forward, leading to a generalization of the relative Hodge-Tate spectral sequence. If time permits, I will give a brief overview of an ongoing project with Michel Gros and Takeshi Tsuji, aimed at establishing a more robust framework for achieving broader functoriality results of the p-adic Simpson correspondence, by both proper push forward and pullback.
2024年02月21日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Jens Niklas Eberhardt 氏 (University of Bonn)
K-motives and Local Langlands (English)
Jens Niklas Eberhardt 氏 (University of Bonn)
K-motives and Local Langlands (English)
[ 講演概要 ]
In this talk, we construct a geometric realisation of the category of representations of the affine Hecke algebra. For this, we introduce a formalism of K-theoretic sheaves (called K-motives) on stacks. The affine Hecke algebra arises from the K-theory of the Steinberg stack, and we explain how to “category” using K-motives.
Lastly, we briefly discuss the relation to the local Langlands program.
In this talk, we construct a geometric realisation of the category of representations of the affine Hecke algebra. For this, we introduce a formalism of K-theoretic sheaves (called K-motives) on stacks. The affine Hecke algebra arises from the K-theory of the Steinberg stack, and we explain how to “category” using K-motives.
Lastly, we briefly discuss the relation to the local Langlands program.
2024年01月24日(水)
17:00-18:00 数理科学研究科棟(駒場) 056号室
Yong Suk Moon 氏 (BIMSA)
Purity for p-adic Galois representations (English)
Yong Suk Moon 氏 (BIMSA)
Purity for p-adic Galois representations (English)
[ 講演概要 ]
Given a smooth p-adic formal scheme, Tsuji proved a purity result for crystalline local systems on its generic fiber. In this talk, we will discuss a generalization for log-crystalline local systems on the generic fiber of a semistable p-adic formal scheme. This is based on a joint work with Du, Liu, and Shimizu.
Given a smooth p-adic formal scheme, Tsuji proved a purity result for crystalline local systems on its generic fiber. In this talk, we will discuss a generalization for log-crystalline local systems on the generic fiber of a semistable p-adic formal scheme. This is based on a joint work with Du, Liu, and Shimizu.
2024年01月10日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
谷田川友里 氏 (東京工業大学)
階数1の層の特性サイクルと部分的に対数的な特性サイクル (Japanese)
谷田川友里 氏 (東京工業大学)
階数1の層の特性サイクルと部分的に対数的な特性サイクル (Japanese)
[ 講演概要 ]
完全体上なめらかな代数多様体上の階数1の層に対して定義されるいくつかの特性サイクルの関係について考える。この特性サイクルのうちの一つはBeilinson-斎藤により構成可能層に対して消失輪体を用いて定義されるものである。
講演では、まず、加藤による対数的な特性サイクルの類似として、階数1の層に対し、対数的および非対数的な分岐理論を用いて部分的に対数的な特性サイクルを構成する。その後、部分的に対数的な特性サイクルを導入する利点や性質、特性サイクルとの関係についてわかったことを紹介する。
完全体上なめらかな代数多様体上の階数1の層に対して定義されるいくつかの特性サイクルの関係について考える。この特性サイクルのうちの一つはBeilinson-斎藤により構成可能層に対して消失輪体を用いて定義されるものである。
講演では、まず、加藤による対数的な特性サイクルの類似として、階数1の層に対し、対数的および非対数的な分岐理論を用いて部分的に対数的な特性サイクルを構成する。その後、部分的に対数的な特性サイクルを導入する利点や性質、特性サイクルとの関係についてわかったことを紹介する。
2023年12月20日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Jinhyun Park 氏 (KAIST)
Accessing the big de Rham-Witt complex via algebraic cycles with a vanishing condition (English)
Jinhyun Park 氏 (KAIST)
Accessing the big de Rham-Witt complex via algebraic cycles with a vanishing condition (English)
[ 講演概要 ]
The big de Rham-Witt complexes of certain good rings over a field are known to admit certain motivic descriptions, namely via cycles with a modulus condition, e.g. additive higher Chow groups. This allowed us to define the trace maps on the de Rham-Witt forms in geometric terms, for instance.
Inspired by a lemma of Kato-Saito on the class field theory and Milnor K-groups, in this talk I would introduce a recent attempt in progress, where a version of “vanishing algebraic cycles” is defined over the formal power series k[[t]]. Using these cycles, I would sketch an alternative cycle-theoretic description of the big de Rham-Witt forms.
The big de Rham-Witt complexes of certain good rings over a field are known to admit certain motivic descriptions, namely via cycles with a modulus condition, e.g. additive higher Chow groups. This allowed us to define the trace maps on the de Rham-Witt forms in geometric terms, for instance.
Inspired by a lemma of Kato-Saito on the class field theory and Milnor K-groups, in this talk I would introduce a recent attempt in progress, where a version of “vanishing algebraic cycles” is defined over the formal power series k[[t]]. Using these cycles, I would sketch an alternative cycle-theoretic description of the big de Rham-Witt forms.
2023年11月22日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
山崎隆雄 氏 (中央大学)
曲面のねじれ双有理モチーフ不変量と不分岐コホモロジー (Japanese)
山崎隆雄 氏 (中央大学)
曲面のねじれ双有理モチーフ不変量と不分岐コホモロジー (Japanese)
[ 講演概要 ]
対角線の分解を許容する曲面の捻じれ双有理モチーフについて，不分岐コホモロジーが普遍的な不変量を与えるという結果について講演する．整係数Hodge予想への新たな反例についても触れる．また，正標数では単純な類似が成立し得ないことを論じる．（佐藤周友氏との共同研究）
対角線の分解を許容する曲面の捻じれ双有理モチーフについて，不分岐コホモロジーが普遍的な不変量を与えるという結果について講演する．整係数Hodge予想への新たな反例についても触れる．また，正標数では単純な類似が成立し得ないことを論じる．（佐藤周友氏との共同研究）
