## Number Theory Seminar

Date, time & place Wednesday 17:00 - 18:00 056Room #056 (Graduate School of Math. Sci. Bldg.) Naoki Imai, Yoichi Mieda

Seminar information archive

### 2013/11/13

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

### 2013/10/30

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

### 2013/10/16

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

### 2013/07/24

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

### 2013/06/26

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

### 2013/06/19

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

### 2013/06/12

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

### 2013/05/29

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

### 2013/05/15

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

### 2013/04/10

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

### 2013/01/16

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

### 2012/12/19

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

### 2012/12/12

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

### 2012/11/14

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

### 2012/07/18

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

### 2012/07/04

16:40-17:40   Room #056 (Graduate School of Math. Sci. Bldg.)
Patrick Forré (University of Tokyo)
A cohomological Hasse principle of varieties over higher local fields and applications to higher dimensional class field theory (ENGLISH)
[ Abstract ]
In this talk I will give an overview of the necessary tools for a description of the class field theory of varieties over higher local fields developed by sevaral mathematicians. On this I will motivate the importance of the proposal and verification of a cohomological Hasse principle for varieties over higher local fields, a generalization of Kato's conjectures, and sketch the recent progress on this.

### 2012/06/20

16:40-17:40   Room #056 (Graduate School of Math. Sci. Bldg.)
Kazuki Tokimoto (University of Tokyo)
On the reduction modulo p of representations of a quaternion
division algebra over a p-adic field (JAPANESE)
[ Abstract ]
The p-adic Langlands correspondence and the mod p Langlands correspondence for GL_2(Q_p) are known to be compatible with the reduction modulo p in many cases.
In this talk, we examine whether a similar compatibility exists for the composition of the local Langlands correspondence and the local Jacquet-Langlands correspondence.
The simplest case has already been treated by Vign¥'eras. We deal with more cases.

### 2012/06/13

16:40-17:40   Room #056 (Graduate School of Math. Sci. Bldg.)
Tomoki Mihara (University of Tokyo)
Singular homologies of non-Archimedean analytic spaces and integrals along cycles (JAPANESE)

### 2012/05/30

16:40-17:40   Room #056 (Graduate School of Math. Sci. Bldg.)
Valentina Di Proietto (University of Tokyo)
Kernel of the monodromy operator for semistable curves (ENGLISH)
[ Abstract ]
For a semistable curve, we study the action of the monodromy operator on the first log-crystalline cohomology group. In particular we examine the relation between the kernel of the monodromy operator and the first rigid cohomology group, in the case of trivial coefficients, giving a new proof of a theorem of B. Chiarellotto and in the case of certain unipotent F-isocrystals as coefficients.
This is a joint work in progress with B. Chiarellotto, R. Coleman and A. Iovita.

### 2012/05/23

16:40-17:40   Room #056 (Graduate School of Math. Sci. Bldg.)
Kentaro Mitsui (University of Tokyo)
Simply connected elliptic surfaces (JAPANESE)
[ Abstract ]
We characterize simply connected elliptic surfaces by their singular fibers in any characteristic case. To this end, we study orbifolds of curves, local canonical bundle formula, and resolutions of multiple fibers. The result was known for the complex analytic case. Our method can be applied to the rigid analytic case.

### 2012/05/16

16:40-17:40   Room #002 (Graduate School of Math. Sci. Bldg.)
Naoya Umezaki (University of Tokyo)
On uniform bound of the maximal subgroup of the inertia group acting unipotently on $¥ell$-adic cohomology (JAPANESE)
[ Abstract ]
For a smooth projective variety over a local field,
the action of the inertia group on the $¥ell$-adic cohomology group is
unipotent if it is restricted to some open subgroup.
In this talk, we give a uniform bound of the index of the maximal open
subgroup satisfying this property.
This bound depends only on the Betti numbers of $X$ and certain Chern
numbers depending on a projective embedding of $X$.

### 2012/04/18

16:40-17:40   Room #056 (Graduate School of Math. Sci. Bldg.)
Alan Lauder (University of Oxford)
Explicit constructions of rational points on elliptic curves (ENGLISH)
[ Abstract ]
I will present an algorithm for computing certain special
values of p-adic L-functions, and discuss an application to
the efficient construction of rational points on elliptic curves.