## 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

### 2011/12/08

18:30-19:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Gerd Faltings (Max Planck Institute for Mathematics, Bonn)
Nonabelian p-adic Hodge theory and Frobenius (ENGLISH)
[ Abstract ]
Some time ago, I constructed a relation between Higgs-bundles and p-adic etale sheaves, on curves over a p-adic field. This corresponds (say in the abelian case) to a Hodge-Tate picture. In the lecture I try to explain one way to introduce Frobenius into the theory. We do not get a complete theory but at least can treat p-adic sheaves close to trivial.

(本講演は「東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とIHESとの双方向同時中継で行います。)

### 2011/11/09

18:00-19:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Atsushi Shiho (University of Tokyo)
On extension and restriction of overconvergent isocrystals (ENGLISH)
[ Abstract ]
First we explain two theorems concerning (log) extension of overconvergent isocrystals. One is a p-adic analogue of the theorem of logarithmic extension of regular integrable connections, and the other is a p-adic analogue of Zariski-Nagata purity. Next we explain a theorem which says that we can check certain property of overconvergent isocrystals by restricting them to curves.

(本講演は「東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とIHESとの双方向同時中継で行います。)

### 2011/11/02

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Kensaku Kinjo (University of Tokyo)
Hypergeometric series and arithmetic-geometric mean over 2-adic fields (JAPANESE)
[ Abstract ]
Dwork proved that the Gaussian hypergeometric function on p-adic numbers
can be extended to a function which takes values of the unit roots of
ordinary elliptic curves over a finite field of characteristic p>2.
We present an analogous theory in the case p=2.
As an application, we give a relation between the canonical lift
and the unit root of an elliptic curve over a finite field of
characteristic 2
by using the 2-adic arithmetic-geometric mean.

### 2011/10/19

17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Andrei Suslin (Northwestern University)
K_2 of the biquaternion algebra (ENGLISH)
[ Reference URL ]
http://www.ihes.fr/~abbes/SGA/suslin.pdf

### 2011/07/27

16:00-18:15   Room #123 (Graduate School of Math. Sci. Bldg.)
Takeshi Saito (University of Tokyo) 16:00-17:00
Discriminants and determinant of a hypersurface of even dimension (ENGLISH)
[ Abstract ]
The determinant of the cohomology of a smooth hypersurface
of even dimension as a quadratic character of the absolute
Galois group is computed by the discriminant of the de Rham
cohomology. They are also computed by the discriminant of a
defining polynomial. We determine the sign involved by testing
the formula for the Fermat hypersurfaces.
This is a joint work with J-P. Serre.
Dennis Eriksson (University of Gothenburg) 17:15-18:15
Multiplicities of discriminants (ENGLISH)
[ Abstract ]
The discriminant of a homogenous polynomial is another homogenous
polynomial in the coefficients of the polynomial, which is zero
if and only if the corresponding hypersurface is singular. In
case the coefficients are in a discrete valuation ring, the
order of the discriminant (if non-zero) measures the bad
reduction. We give some new results on this order, and in
particular tie it to Bloch's conjecture/the Kato-T.Saito formula
on equality of localized Chern classes and Artin conductors. We
can precisely compute all the numbers in the case of ternary
forms, giving a partial generalization of Ogg's formula for
elliptic curves.

### 2011/06/15

17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Tomoyuki Abe (IPMU)
Product formula for $p$-adic epsilon factors (ENGLISH)
[ Abstract ]
I would like to talk about my recent work jointly with A. Marmora on a product formula for $p$-adic epsilon factors. In 80's Deligne conjectured that a constant appearing in the functional equation of $L$-function of $\\ell$-adic lisse sheaf can be written by means of local contributions, and proved some particular cases. This conjecture was proven later by Laumon, and was used in the Lafforgue's proof of the Langlands' program for functional filed case. In my talk, I would like to prove a $p$-adic analog of this product formula.

### 2011/06/08

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Yuichi Hirano (University of Tokyo)
Congruences of modular forms and the Iwasawa λ-invariants (JAPANESE)

### 2011/05/25

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Yuya Matsumoto (University of Tokyo)
On good reduction of some K3 surfaces (JAPANESE)

### 2011/05/18

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Masaki Nishimoto (University of Tokyo)
On the linear independence of values of some Dirichlet series (JAPANESE)

### 2011/05/11

17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Michel Raynaud (Universite Paris-Sud)
Permanence following Temkin (ENGLISH)
[ Abstract ]
When one proceeds to a specialization, the good properties of algebraic equations may be destroyed. Starting with a bad specialization, one can try to improve it by performing modifications under control. If, at the end of the process, the initial good properties are preserved, one speaks of permanence. I shall give old and new examples of permanence. The new one concerns the relative semi-stable reduction of curves recently proved by Temkin.

### 2011/04/27

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Yuuki Takai (University of Tokyo)
An analogue of Sturm's theorem for Hilbert modular forms (JAPANESE)

### 2011/02/10

11:00-12:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Joseph Ayoub (University of Zurich)
The motivic Galois group and periods of algebraic varieties (ENGLISH)
[ Abstract ]
We give a construction of the motivic Galois group of $\\Q$ and explain the conjectural link with the ring of periods of algebraic varieties. Then we introduce the ring of formal periods and explain how the conjectural link with the motivic Galois group can be realized for them.

### 2011/01/26

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Shinichi Kobayashi (Tohoku University)
The p-adic Gross-Zagier formula for elliptic curves at supersingular primes (JAPANESE)
[ Abstract ]
The p-adic Gross-Zagier formula is a formula relating the derivative of the p-adic L-function of elliptic curves to the p-adic height of Heegner points. For a good ordinary prime p, the formula is proved by B. Perrin-Riou more than 20 years ago. Recently, the speaker proved it for a supersingular prime p. In this talk, he explains the proof.

### 2011/01/12

16:30-18:45   Room #056 (Graduate School of Math. Sci. Bldg.)
Zhonghua Li (University of Tokyo) 16:30-17:30
On regularized double shuffle relation for multiple zeta values (ENGLISH)
[ Abstract ]
Multiple zeta values(MZVs) are natural generalizations of Riemann zeta values. There are many rational relations among MZVs. It is conjectured that the regularized double shuffle relations contian all rational relations of MZVs. So other rational relations should be deduced from regularized dhouble shuffle relations. In this talk, we discuss some results on this problem. We define the gamma series accociated to elements satisfying regularized double shuffle relations and give some properties. Moreover we show that the Ohno-Zagier relations can be deduced from regularized double shuffle relations.
Dan Yasaki (North Carolina University) 17:45-18:45
Spines with View Toward Modular Forms (ENGLISH)
[ Abstract ]
The study of an arithmetic group is often aided by the fact that it acts naturally on a nice topological object. One can then use topological or geometric techniques to try to recover arithmetic data. For example, one often studies SL_2(Z) in terms of
its action on the upper half plane. In this talk, we will examine spines, which are the smallest" such spaces for a given arithmetic group. On overview of some known theoretical results and explicit computations will be given.

### 2010/12/22

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Takashi Hara (University of Tokyo)
Inductive construction of the p-adic zeta functions for non-commutative
p-extensions of totally real fields with exponent p (JAPANESE)
[ Abstract ]
We will discuss how to construct p-adic zeta functions and verify
the main conjecture in special cases in non-commutative Iwasawa theory
for totally real number fields.

The non-commutative Iwasawa main conjecture for totally real number
fields has been verified in special cases by Kazuya Kato,
Mahesh Kakde and the speaker by patching method of p-adic zeta functions'
introduced by David Burns and Kazuya Kato (Jurgen Ritter and Alfred Weiss
have also constructed the successful example of the main conjecture
under somewhat different formulations).

In this talk we will explain that we can prove the main conjecture
for cases where the Galois group is isomorphic
to the direct product of the ring of p-adic integer and a finite p-group
of exponent p by utilizing Burns-Kato's method and inductive arguments.

Finally we remark that in 2010 Ritter-Weiss and Kakde independently
justified the non-commutative main conjecture
for totally real number fields under general settings.

### 2010/12/01

16:30-18:45   Room #056 (Graduate School of Math. Sci. Bldg.)
Yuichiro Hoshi (RIMS, Kyoto University) 16:30-17:30
On a problem of Matsumoto and Tamagawa concerning monodromic fullness of hyperbolic curves (JAPANESE)
[ Abstract ]
In this talk, we will discuss the following problem posed by Makoto Matsumoto and Akio Tamagawa concerning monodromic fullness of hyperbolic curves.

For a hyperbolic curve X over a number field, are the following three conditions equivalent?
(A) For any prime number l, X is quasi-l-monodromically full.
(B) There exists a prime number l such that X is l-monodromically full.
(C) X is l-monodromically full for all but finitely many prime numbers l.

The property of being (quasi-)monodromically full may be regarded as an analogue for hyperbolic curves of the property of not admitting complex multiplication for elliptic curves, and the above equivalence may be regarded as an analogue for hyperbolic curves of the following result concerning the Galois representation on the Tate module of an elliptic curve over a number field proven by Jean-Pierre Serre.

For an elliptic curve E over a number field, the following four conditions are equivalent:
(0) E does not admit complex multiplication.
(1) For any prime number l, the image of the l-adic Galois representation associated to E is open.
(2) There exists a prime number l such that the l-adic Galois representation associated to E is surjective.
(3) The l-adic Galois representation associated to E is surjective for all but finitely many prime numbers l.

In this talk, I will present some results concerning the above problem in the case where the given hyperbolic curve is of genus zero. In particular, I will give an example of a hyperbolic curve of type (0,4) over a number field which satisfies condition (C) but does not satisfy condition (A).
Marco Garuti (University of Padova) 17:45-18:45
Galois theory for schemes (ENGLISH)
[ Abstract ]
We discuss some aspects of finite group scheme actions: the Galois correspondence and the notion of Galois closure.

### 2010/11/17

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Shin Harase (University of Tokyo)
Fast lattice reduction for F_2-linear pseudorandom number generators (JAPANESE)

### 2010/10/06

16:30-17:30   Room #117 (Graduate School of Math. Sci. Bldg.)
Hélène Esnault (Universität Duisburg-Essen)
Finite group actions on the affine space (ENGLISH)
[ Abstract ]
If $G$ is a finite $\\ell$-group acting on an affine space $\\A^n$ over a
finite field $K$ of cardinality prime to $\\ell$, Serre shows that there
exists a rational fixed point. We generalize this to the case where $K$ is a
henselian discretely valued field of characteristic zero with algebraically
closed residue field and with residue characteristic different from $\\ell$.
We also treat the case where the residue field is finite of cardinality $q$
such that $\\ell$ divides $q-1$. To this aim, we study group actions on weak
N\\'eron models.
(Joint work with Johannes Nicaise)

### 2010/07/07

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Takahiro Tsushima (University of Tokyo)
On the stable reduction of $X_0(p^4)$ (JAPANESE)

### 2010/06/16

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Luc Illusie (Universite de Paris-Sud)
Vanishing theorems revisited, after K.-W. Lan and J. Suh (ENGLISH)
[ Abstract ]
Let k be an algebraically closed field of characteristic p and X,
Y proper, smooth k-schemes. J. Suh has proved a vanishing theorem of Kollar
type for certain nef and big line bundles L on Y and morphisms f : X -> Y
having semistable reduction along a divisor with simple normal crossings. It
holds both if p = 0 and if p > 0 modulo some additional liftability mod p^2
and dimension assumptions, and generalizes vanishing theorems of Esnault-
Viehweg and of mine. I'll give an outline of the proof and sketch some
applications, due to K.-W. Lan and J. Suh, to the cohomology of certain
automorphic bundles arising from PEL type Shimura varieties.

### 2010/06/09

16:15-17:15   Room #052 (Graduate School of Math. Sci. Bldg.)
Richard Hain (Duke University)
Universal mixed elliptic motives (ENGLISH)
[ Abstract ]
This is joint work with Makoto Matsumoto. A mixed elliptic
motive is a mixed motive (MHS, Galois representation, etc) whose
weight graded quotients are Tate twists of symmetric powers of the the
motive of elliptic curve. A universal mixed elliptic motive is an
object that can be specialized to a mixed elliptic motive for any
elliptic curve and whose specialization to the nodal cubic is a mixed
Tate motive. Universal mixed elliptic motives form a tannakian
category. In this talk I will define universal mixed elliptic motives,
give some fundamental examples, and explain what we know about the
fundamental group of this category. The "geometric part" of this group
is an extension of SL_2 by a prounipotent group that is generated by
Eisenstein series and which has a family of relations for each cusp
form. Although these relations are not known, we have a very good idea
of what they are, thanks to work of Aaron Pollack, who determined
relations between the generators in a very large representation of
this group.

### 2010/06/09

17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Fabrice Orgogozo (CNRS, École polytechnique)
Constructibilité uniforme des images directes supérieures en
cohomologie étale
(ENGLISH)
[ Abstract ]
Motivé par une remarque de N. Katz sur le lien entre la
torsion de la Z_ℓ-cohomologie étale et les ultraproduits de groupes de
F_ℓ-cohomologie, nous démontrons un théorème d'uniformité en ℓ pour la
constructibilité des images directes supérieures entre schémas de type fini
sur un trait excellent. (Un tel théorème avait été considéré par
O. Gabber il y a plusieurs années déjà.)
La méthode est maintenant classique : on utilise des
théorèmes de A. J. de Jong et un peu de log-géométrie.

(This lecture is held as Arithmetic Geometry Seminar Tokyo-Paris' and it is transmitted from IHES by the internet.)

### 2010/06/02

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Ryoko Tomiyasu (KEK)
On some algebraic properties of CM-types of CM-fields and their
reflex fields (JAPANESE)
[ Abstract ]
Shimura and Taniyama proved in their theory of complex
multiplication that the moduli of abelian varieties of a CM-type and their
torsion points generate an abelian extension, not of the field of complex
multiplication, but of a reflex field of the field. In this talk, I
introduce some algebraic properties of CM-types, half norm maps that might
shed new light on reflex fields.

For a CM-field $K$ and its Galois closure $K^c$ over the rational field $Q$,
there is a canonical embedding of $Gal (K^c/Q)$ into $(Z/2Z)^n \\rtimes S_n$.
Using properties of the embedding, a set of CM-types $\\Phi$ of $K$ and their
dual CM-types $(K, \\Phi)$ is equipped with a combinatorial structure. This
makes it much easier to handle a whole set of CM-types than an individual
CM-type.

I present a theorem that shows the combinatorial structure of the dual
CM-types is isomorphic to that of a Pfister form.

### 2010/05/12

17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Makoto Matsumoto (University of Tokyo)
Differences between
Galois representations in outer-automorphisms
of the fundamental groups and those in automorphisms, implied by
topology of moduli spaces (ENGLISH)
[ Abstract ]
Fix a prime l. Let C be a proper smooth geometrically connected curve over a number ﬁeld K, and x be its closed point. Let Π denote the pro-l completion of the geometric fundamental group of C with geometric base point over x. We have two non-abelian Galois representations:

ρA : Galk(x) → Aut(Π),ρO : GalK → Out(Π).

Our question is: in the natural inclusion Ker(ρA) ⊂ Ker(ρO) ∩ Galk(x), whether the equality holds or not. Theorem: Assume that g ≥ 3, l divides 2g -2. Then, there are inﬁnitely many pairs (C, K) with the following property. If l does not divide the extension degree [k(x): K], then Ker(ρA) = (Ker(ρO) ∩ Galk(x)) holds.

This is in contrast to the case of the projective line minus three points and its canonical tangential base points, where the equality holds (a result of Deligne and Ihara).

There are two ingredients in the proof: (1) Galois representations often contain the image of the geometric monodromy (namely, the mapping class group) [M-Tamagawa 2000] (2) A topological result [S. Morita 98] [Hain-Reed 2000] on the cohomological obstruction of lifting the outer action of the mapping class group to automorphisms.

(This lecture is held as `Arithmetic Geometry Seminar Tokyo-Paris' and it is transmitted to IHES by the internet.)

### 2010/04/14

17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Gerard Laumon (CNRS, Universite Paris XI - Orsay)
The cohomological weighted fundamental lemma
[ Abstract ]
Using the Hitchin fibration, Ngo Bao Chau has proved the Langlands-Shelstad fundamental lemma. In a joint work with Pierre-Henri Chaudouard, we have extended Ngo's proof to obtain the weighted fundamental lemma which had been conjectured by Arthur. In the talk, I would like to present our main cohomological result.

(本講演は「東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とIHESとの双方向同時中継で行います。)