## Number Theory Seminar

Seminar information archive ～12/10｜Next seminar｜Future seminars 12/11～

Date, time & place | Wednesday 17:00 - 18:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
---|---|

Organizer(s) | Naoki Imai, Shane Kelly |

**Seminar information archive**

### 2014/02/05

17:10-18:10 Room #002 (Graduate School of Math. Sci. Bldg.)

The Franke filtration of spaces of automorphic forms (ENGLISH)

**Neven Grbac**(University of Rijeka)The Franke filtration of spaces of automorphic forms (ENGLISH)

[ Abstract ]

The Franke filtration is a filtration of the space of all adelic automorphic forms on a reductive group defined over a number field. The filtration steps can be described as certain induced representations, which has applications to the study of Eisenstein cohomology. In this talk, we shall describe the Franke filtration in general, give several examples, and explain its connection to cohomology.

The Franke filtration is a filtration of the space of all adelic automorphic forms on a reductive group defined over a number field. The filtration steps can be described as certain induced representations, which has applications to the study of Eisenstein cohomology. In this talk, we shall describe the Franke filtration in general, give several examples, and explain its connection to cohomology.

### 2014/01/24

16:40-18:50 Room #056 (Graduate School of Math. Sci. Bldg.)

An approach to p-adic Hodge theory over number fields (ENGLISH)

Canonical lifts of norm fields and applications (ENGLISH)

**Christopher Davis**(University of Copenhagen) 16:40-17:40An approach to p-adic Hodge theory over number fields (ENGLISH)

[ Abstract ]

As motivation from classical Hodge theory, we will first compare singular cohomology and (algebraic) de Rham cohomology for a complex analytic variety. We will also describe a sense in which this comparison does not have a natural analogue over the real numbers. We think of the complex numbers as a "big" ring which is necessary for the comparison isomorphism to work. In the p-adic setting, the analogous study is known as p-adic Hodge theory, and the "big" rings there are even bigger. There are many approaches to p-adic Hodge theory, and we will introduce one tool in particular: (phi, Gamma)-modules. The goal of this talk is to describe a preliminary attempt to find an analogue of this theory (and analogues of its "big" rings) which makes sense over number fields (rather than p-adic fields). This is joint work with Kiran Kedlaya.

As motivation from classical Hodge theory, we will first compare singular cohomology and (algebraic) de Rham cohomology for a complex analytic variety. We will also describe a sense in which this comparison does not have a natural analogue over the real numbers. We think of the complex numbers as a "big" ring which is necessary for the comparison isomorphism to work. In the p-adic setting, the analogous study is known as p-adic Hodge theory, and the "big" rings there are even bigger. There are many approaches to p-adic Hodge theory, and we will introduce one tool in particular: (phi, Gamma)-modules. The goal of this talk is to describe a preliminary attempt to find an analogue of this theory (and analogues of its "big" rings) which makes sense over number fields (rather than p-adic fields). This is joint work with Kiran Kedlaya.

**Bryden Cais**(University of Arizona) 17:50-18:50Canonical lifts of norm fields and applications (ENGLISH)

[ Abstract ]

In this talk, we begin by outlining the Fontaine-Wintenberger theory of norm fields and explain its application to the classification of p-adic Galois representations on F_p-vector spaces. In order to lift this to a classification of p-adic representations on Z_p-modules, it is necessary to lift the characteristic p norm field constructions of Fontaine-Wintenberger to characteristic zero. We will explain how to canonically perform such lifting in many interesting cases, as well as applications to generalizing a theorem of Kisin on the restriction of crystalline p-adic Galois representations. This is joint work with Christopher Davis.

In this talk, we begin by outlining the Fontaine-Wintenberger theory of norm fields and explain its application to the classification of p-adic Galois representations on F_p-vector spaces. In order to lift this to a classification of p-adic representations on Z_p-modules, it is necessary to lift the characteristic p norm field constructions of Fontaine-Wintenberger to characteristic zero. We will explain how to canonically perform such lifting in many interesting cases, as well as applications to generalizing a theorem of Kisin on the restriction of crystalline p-adic Galois representations. This is joint work with Christopher Davis.

### 2014/01/22

18:00-19:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Riemann-Hilbert correspondence for irregular holonomic D-modules (ENGLISH)

**Masaki Kashiwara**(RIMS, Kyoto University)Riemann-Hilbert correspondence for irregular holonomic D-modules (ENGLISH)

[ Abstract ]

The classical Riemann-Hilbert correspondence establishes an equivalence between the derived category of regular holonomic D-modules and the derived category of constructible sheaves. Recently, I, with Andrea D'Agnolo, proved a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular (arXiv:1311.2374). In this correspondence, we have to replace the derived category of constructible sheaves with a full subcategory of ind-sheaves on the product of the base space and the real projective line. The construction is therefore based on the theory of ind-sheaves by Kashiwara-Schapira, and also it is influenced by Tamarkin's work. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Takuro Mochizuki and Kiran Kedlaya.

The classical Riemann-Hilbert correspondence establishes an equivalence between the derived category of regular holonomic D-modules and the derived category of constructible sheaves. Recently, I, with Andrea D'Agnolo, proved a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular (arXiv:1311.2374). In this correspondence, we have to replace the derived category of constructible sheaves with a full subcategory of ind-sheaves on the product of the base space and the real projective line. The construction is therefore based on the theory of ind-sheaves by Kashiwara-Schapira, and also it is influenced by Tamarkin's work. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Takuro Mochizuki and Kiran Kedlaya.

### 2014/01/15

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Special values of zeta-functions of schemes (ENGLISH)

**Stephen Lichtenbaum**(Brown University)Special values of zeta-functions of schemes (ENGLISH)

[ Abstract ]

We will give conjectured formulas giving the behavior of the

seta-function of regular schemes projective and flat over Spec Z at

non-positive integers in terms of Weil-etale cohomology. We will also

explain the conjectured relationship of Weil-etale cohomology to etale

cohomology, which makes it possible to express these formulas also in terms

of etale cohomology.

We will give conjectured formulas giving the behavior of the

seta-function of regular schemes projective and flat over Spec Z at

non-positive integers in terms of Weil-etale cohomology. We will also

explain the conjectured relationship of Weil-etale cohomology to etale

cohomology, which makes it possible to express these formulas also in terms

of etale cohomology.

### 2014/01/08

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Roots of the discriminant of an elliptic curves and its torsion points (JAPANESE)

**Sho Yoshikawa**(University of Tokyo)Roots of the discriminant of an elliptic curves and its torsion points (JAPANESE)

[ Abstract ]

We give an explicit and intrinsic description of (the torsor defined by the 12th roots of) the discriminant of an elliptic curve using the group of its 12-torsion points and the Weil pairing. As an application, we extend a result of Coates (which deals with the characteristic 0 case) to the case where the characteristic of the base field is not 2 or 3. This is a joint work with Kohei Fukuda.

We give an explicit and intrinsic description of (the torsor defined by the 12th roots of) the discriminant of an elliptic curve using the group of its 12-torsion points and the Weil pairing. As an application, we extend a result of Coates (which deals with the characteristic 0 case) to the case where the characteristic of the base field is not 2 or 3. This is a joint work with Kohei Fukuda.

### 2013/12/18

18:00-19:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Heights of motives (ENGLISH)

**Kazuya Kato**(University of Chicago)Heights of motives (ENGLISH)

[ Abstract ]

The height of a rational number a/b (a, b integers which are coprime) is defined as max(|a|, |b|). A rational number with small (resp. big) height is a simple (resp. complicated) number. Though the notion height is so naive, height has played fundamental roles in number theory. There are important variants of this notion. In 1983, when Faltings proved Mordell conjecture, Faltings first proved Tate conjecture for abelian variaties by defining heights of abelian varieties, and then he deduced Mordell conjecture from the latter conjecture. I explain that his height of an abelian variety is generalized to the height of a motive. This generalization of height is related to open problems in number theory. If we can prove finiteness of the number of motives of bounded heights, we can prove important conjectures in number theory such as general Tate conjecture and Mordell-Weil type conjectures in many cases.

The height of a rational number a/b (a, b integers which are coprime) is defined as max(|a|, |b|). A rational number with small (resp. big) height is a simple (resp. complicated) number. Though the notion height is so naive, height has played fundamental roles in number theory. There are important variants of this notion. In 1983, when Faltings proved Mordell conjecture, Faltings first proved Tate conjecture for abelian variaties by defining heights of abelian varieties, and then he deduced Mordell conjecture from the latter conjecture. I explain that his height of an abelian variety is generalized to the height of a motive. This generalization of height is related to open problems in number theory. If we can prove finiteness of the number of motives of bounded heights, we can prove important conjectures in number theory such as general Tate conjecture and Mordell-Weil type conjectures in many cases.

### 2013/11/20

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

On the homotopy exact sequence for the logarithmic de Rham fundamental group (ENGLISH)

**Valentina Di Proietto**(The University of Tokyo)On the homotopy exact sequence for the logarithmic de Rham fundamental group (ENGLISH)

[ Abstract ]

Let K be a field of characteristic 0 and let X* be a quasi-projective simple normal crossing log variety over the log point K* associated to K. We construct a log de Rham version of the homotopy sequence \\pi_1(X*/K*)-->\\pi_1(X*/K)--\\pi_1(K*/K)-->1 and prove that it is exact. Moreover we show the injectivity of the first map for certain quotients of the groups. Our proofs are purely algebraic. This is a joint work with A. Shiho.

Let K be a field of characteristic 0 and let X* be a quasi-projective simple normal crossing log variety over the log point K* associated to K. We construct a log de Rham version of the homotopy sequence \\pi_1(X*/K*)-->\\pi_1(X*/K)--\\pi_1(K*/K)-->1 and prove that it is exact. Moreover we show the injectivity of the first map for certain quotients of the groups. Our proofs are purely algebraic. This is a joint work with A. Shiho.

### 2013/11/13

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Goren-Oort stratification and Tate cycles on Hilbert modular varieties (ENGLISH)

**Yichao Tian**(Morningside Center for Mathematics)Goren-Oort stratification and Tate cycles on Hilbert modular varieties (ENGLISH)

[ Abstract ]

Let B be a quaternionic algebra over a totally real field F, and p be a prime at least 3 unramified in F. We consider a Shimura variety X associated to B^* of level prime to p. A generalization of Deligne-Carayol's "modèle étrange" allows us to define an integral model for X. We will then define a Goren-Oort stratification on the characteristic p fiber of X, and show that each closed Goren-Oort stratum is an iterated P^1-fibration over another quaternionic Shimura variety in characteristic p. Now suppose that [F:Q] is even and that p is inert in F. An iteration of this construction gives rise to many algebraic cycles of middle codimension on the characteristic p fibre of Hilbert modular varieties of prime-to-p level. We show that the cohomological classes of these cycles generate a large subspace of the Tate cycles, which, in some special cases, coincides with the prediction of the Tate conjecture for the Hilbert modular variety over finite fields. This is a joint work with Liang Xiao.

Let B be a quaternionic algebra over a totally real field F, and p be a prime at least 3 unramified in F. We consider a Shimura variety X associated to B^* of level prime to p. A generalization of Deligne-Carayol's "modèle étrange" allows us to define an integral model for X. We will then define a Goren-Oort stratification on the characteristic p fiber of X, and show that each closed Goren-Oort stratum is an iterated P^1-fibration over another quaternionic Shimura variety in characteristic p. Now suppose that [F:Q] is even and that p is inert in F. An iteration of this construction gives rise to many algebraic cycles of middle codimension on the characteristic p fibre of Hilbert modular varieties of prime-to-p level. We show that the cohomological classes of these cycles generate a large subspace of the Tate cycles, which, in some special cases, coincides with the prediction of the Tate conjecture for the Hilbert modular variety over finite fields. This is a joint work with Liang Xiao.

### 2013/10/30

16:40-17:40 Room #002 (Graduate School of Math. Sci. Bldg.)

Explicit integral cocycles on GLn and special values of p-adic partial zeta functions (ENGLISH)

**Pierre Charollois**(Université Paris 6)Explicit integral cocycles on GLn and special values of p-adic partial zeta functions (ENGLISH)

[ Abstract ]

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.

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.

### 2013/10/16

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Shimura varieties with infinite level, and torsion in the cohomology of locally symmetric spaces (ENGLISH)

**Peter Scholze**(Universität Bonn)Shimura varieties with infinite level, and torsion in the cohomology of locally symmetric spaces (ENGLISH)

[ Abstract ]

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.

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.

### 2013/07/24

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

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

**Yasuhiro Terakado**(University of Tokyo)The determinant of a double covering of the projective space of even dimension and the discriminant of the branch locus (JAPANESE)

### 2013/07/10

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

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

**Yuri Yatagawa**(University of Tokyo)On ramification filtration of local fields of equal characteristic (JAPANESE)

### 2013/07/03

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

A general formula for the discriminant of polynomials over $¥mathbb{F}_2$ determining the parity of the number of prime factors

(JAPANESE)

**Takehito Yoshiki**(University of Tokyo)A general formula for the discriminant of polynomials over $¥mathbb{F}_2$ determining the parity of the number of prime factors

(JAPANESE)

[ Abstract ]

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.

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.

### 2013/06/26

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

An explicit construction of point sets with large minimum Dick weight (JAPANESE)

**Kousuke Suzuki**(University of Tokyo)An explicit construction of point sets with large minimum Dick weight (JAPANESE)

[ Abstract ]

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.

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.

### 2013/06/19

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

A p-adic exponential map for the Picard group and its application to curves (JAPANESE)

**Wataru Kai**(University of Tokyo)A p-adic exponential map for the Picard group and its application to curves (JAPANESE)

[ Abstract ]

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 $k$.

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 $k$.

### 2013/06/12

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Hodge index theorem for adelic line bundles (ENGLISH)

**Xinyi Yuan**(University of California, Berkeley)Hodge index theorem for adelic line bundles (ENGLISH)

[ Abstract ]

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.

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.

### 2013/05/29

16:40-17:40 Room #002 (Graduate School of Math. Sci. Bldg.)

On logarithmic nonabelian Hodge theory of higher level in characteristic p (JAPANESE)

**Sachio Ohkawa**(University of Tokyo)On logarithmic nonabelian Hodge theory of higher level in characteristic p (JAPANESE)

[ Abstract ]

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.

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.

### 2013/05/15

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

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

**Hiroyasu Miyazaki**(University of Tokyo)Special values of zeta functions of singular varieties over finite fields via higher chow groups (JAPANESE)

### 2013/04/24

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Good reduction of ramified affinoids in the Lubin-Tate perfectoid space (ENGLISH)

**Naoki Imai**(University of Tokyo)Good reduction of ramified affinoids in the Lubin-Tate perfectoid space (ENGLISH)

[ Abstract ]

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.

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.

### 2013/04/10

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Motivic structure on higher homotopy of non-nilpotent spaces (ENGLISH)

**Deepam Patel**(University of Amsterdam)Motivic structure on higher homotopy of non-nilpotent spaces (ENGLISH)

[ Abstract ]

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.

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.

### 2013/01/16

18:00-19:00 Room #002 (Graduate School of Math. Sci. Bldg.)

On differential modules associated to de Rham representations in the imperfect residue field case (ENGLISH)

**Shun Ohkubo**(University of Tokyo)On differential modules associated to de Rham representations in the imperfect residue field case (ENGLISH)

[ Abstract ]

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.

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.

### 2012/12/19

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

A generalization of Kato's local epsilon conjecture for

(φ, Γ)-modules over the Robba ring (JAPANESE)

**Kentarou Nakamura**(Hokkaido University)A generalization of Kato's local epsilon conjecture for

(φ, Γ)-modules over the Robba ring (JAPANESE)

[ Abstract ]

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.

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.

### 2012/12/12

18:00-19:00 Room #002 (Graduate School of Math. Sci. Bldg.)

The Tate conjecture for K3 surfaces and holomorphic symplectic varieties over finite fields (ENGLISH)

**François Charles**(CNRS & Université de Rennes 1)The Tate conjecture for K3 surfaces and holomorphic symplectic varieties over finite fields (ENGLISH)

[ Abstract ]

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.

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.

### 2012/11/14

18:00-19:00 Room #002 (Graduate School of Math. Sci. Bldg.)

De Rham-Witt complexes with coefficients and rigid cohomology

(ENGLISH)

**Pierre Berthelot**(Université de Rennes 1)De Rham-Witt complexes with coefficients and rigid cohomology

(ENGLISH)

[ Abstract ]

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.

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.

### 2012/07/18

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Voevodsky motives and a theorem of Gabber (ENGLISH)

**Shane Kelly**(Australian National University)Voevodsky motives and a theorem of Gabber (ENGLISH)

[ Abstract ]

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.

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.