## The 18th Number Theory Meeting at Waseda Univ.

### Date

Tuesday, 11th March, 2014 -- Thursday, 13th March, 2014

### Place

2-nd conference room, bldg. 55-N 1F, Nishiwaseda campus Waseda University

### Schedule

date | 1 | 2 | 3 | lunch | 4 | 5 | 6 | 7 | Party |
---|---|---|---|---|---|---|---|---|---|

3/11 (Tue) | 10:00-10:45 | 11:05-11:50 | ~ | lunch | 14:00-14:45 | 15:05-15:50 | 16:10-16:55 | 17:15-18:00 | ~ |

3/12 (Wed) | 10:00-10:45 | 11:05-11:50 | ~ | lunch | 14:00-14:45 | 15:05-15:50 | 16:10-16:55 | ~ | 18:00-20:00 |

3/13 (Thu) | 9:30-10:15 | 10:35-11:20 | 11:40-12:25 | ~ | ~ | ~ | ~ | ~ | ~ |

### Organizers

Keiichi Komatsu (Waseda University),
Kiichiro Hashimoto (Waseda University),

Manabu Ozaki (Waseda University),
Hiroshi Sakata (Waseda University Senior High School)

### Supporting Organizations

JSPS Grant-in-Aid for Scientic Research (C) (24540030,Keiichi Komatsu) and (21540030, Manabu Ozaki).

### PDF file

Program@@ Abstract

## Tuesday, 11th March

### 10:00--10:45 Shuichi Ohtake (Waseda University)

Title :

Orthogonal decompositions of integral trace forms of cyclotomic
fields

Abstract :

Trace form is the symmetric *F*-bilinear form on *K:= F[x]=(f(x))*
defined by *(x,y) ¨ trace _{K/F}(xy)*, where

*f(x)*is a separable polynomial over the field

*F*of characteristic different from two.

If

*f(x)*is an irreducible polynomial over the field of rational numbers

*Q*, then the restriction of the trace form to the ring of integers

*O*of the number field

_{K}*K*defines a symmetric bilinear form over the ring of rational integers

*Z*on

*O*- called the integral trace form of

_{K}*K*.

In this talk, we give orthogonal decompositions of integral trace forms of cyclotomic fields and their canonical forms over the ring of

*p*-adic integers explicitly by using Bezoutian forms.

### 11:05--11:50 Satoshi Fujii (Kanazawa Institute of Technology)

Title :

On Greenberg's conjecture for CM-fields

Abstract :

Let *k/Q* be a finite extension and *p* an
odd prime number.

Let *K/k* be the maximal multiple *Z _{p}*-extension,
and let

*Gal(K/k) [“¯Œ^] Z*.

_{p}^{d}Let

*L/K*be the maximal unramified abelian pro-

*p*extension and

*X*its Galois group.

By a fundamental fact of Iwasawa theory,

*X*is a module over the formal power series ring

*ƒ©*with coefficients in

*Z*of

_{p}*d*-variables.

Then Greenberg conjectured that

*X*is pseudo-null over

*ƒ©*.

In this talk, we show that Greenbergfs conjecture holds under the following four conditions:

1) k is a CM-field such that

*p*splits completely.

2) Leopoldtfs conjecture holds for

*k*and

*p*(e.g.

*k*is imaginary abelian).

3)

*p*does not divide the class number of

*k*,

4)

*ƒÉ*, where

_{p}(k^{+}) = ƒÊ_{p}(k^{+}) = ƒË_{p}(k^{+}) = 0*ƒÉ, ƒÊ*and

*ƒË*denote Iwasawa invariants and

*k*denotes the maximal totally real subfield of

^{+}*k*.

### 14:00 -- 14:45 Manabu Ozaki (Waseda University)

Title :

The Neukirch-Uchida theorem for a certain class of number fields
of infinite degree

Abstract :

I will give a Neukirch-Uchida type theorem
(that is, the isomorphism class of a field is
determined from its absolute Galois group)
for a certain class of number fields of
infinite degree.

### 15:05--15:50 Yuichiro Hoshi (Kyoto University)

Title :

Reconstruction of a Number Field From the Absolute Galois
Group

Abstract :

It follows from the Neukirch-Uchida Theorem that the isomorphism
class of a number field is completely determined by the isomorphism
class of the associated absolute Galois group.

On the other hand, the Neukirch-Uchida Theorem
(as well as its proof) does not give a hfunctorial
grouptheoretic algorithmh for reconstructing the
original number field from the absolute
Galois group.

In this talk, I discuss such a hfunctorial group-theoretic
algorithmh.

### 16:10--16:55 Kenji Sakugawa (Osaka University)

Title :

A control theorem for the torsion Selmer pointed set

Abstract :

Selmer groups are important arithmetic invariants
of Galois representations.

Minhyong Kim defined the Selmer variety which
is a

non-abelian analogue of the
*Q _{p}*-Selmer group.

In this talk, we give a torsion analogue of the Selmer variety.

Then, we establish an analogue of Mazurfs control theorem for this torsion analogue.

### 17:15--18:00 Kentaro Nakamura (Hokkaido University)

Title :

Local ƒÃ-isomorphisms for rank two *p*-adic representations of
Gal((Q_p)^[/Q_p) and a functional equation of Kato's Euler system

Abstract :

Local *ƒÃ*-isomorphisms are conjectural bases
of the determinants of the Galois cohomologies
of *p*-adic representations of
*Gal(Q _{p}^{[}/Q_{p})*
which

*p*-adically interpolate local constants (

*ƒÃ*-constatnts,

*L*-constants, etc.) associated to de Rham representations.

Up to now, such bases have been constructed for rank one case by Kazuya Kato, crystalline case by Benois-Berger, Loeffler-Zerbes-Venjakob, and trianguline case by the speaker.

In this talk, using Colmezfs theory of

*p*-adic Langlands correspondence for

*GL*, we define such bases for (almost) all rank two torsion

_{2}(Q_{p})*p*-adic representations.

We show that our

*ƒÃ*-isomorphisms satisfy the desired interpolation property in many important cases.

As an application, we prove a functional equation of Katofs Euler systems associated to modular forms without any condition at

*p*.

Under the assumption that Katofs Euler systems give the zeta isomorphisms, this functional equation implies Katofs global

*ƒÃ*-conjecture.

## Wednesday, 12th March

### 10:00--10:45 Masao Tsuduki (Sophia University)

Title :

Equidistribution and subconvexity bound related to certain
*L*-values

Abstract :

This is joint work with Singo Sugiyama (Osaka Univ.).

I would like to report our recent refinement on
a spectral equidistribution result in the level
aspect for Satake parameters of holomorphic Hilbert cusp
forms weighted by central *L*-values,
and a bound of quadratic base change *L*-functions
for Hilbert cusp forms with a subconvex exponent in the
weight aspect.

### 11:05--11:50 Hiroki Aoki (Tokyo University of Science)

Title :

Determination of the structure of vector valued Siegel modular
forms by using Jacobi forms

Abstract :

In general, the determination of the structure of
modular forms is difficult, although the dimension
formula is well known.

However, sometimes by using Jacobi forms or
Witt operators, we can easily determine the structure
of some kinds of modular forms.

In this talk, I shall introduce this strategy
and show some examples, including the structure
theorem of vector valued Siegel modular
forms of level *2*.

### 14:00--14:45 Thomas WieberiHeidelberg Universityj

Title :

Geometrically proven structure theorems for vector valued Siegel
modular forms

Abstract :

I shall begin with classical results on vector valued
(cuspidal) Siegel modular forms.

Afterwards, I shall present new structure theorems
for vector valued Siegel modular forms with respect
to Sym^{2} and Igusafs subgroup
*ƒ¡ _{2}[2, 4]*.
They rest on the well known fact that

*ƒ¡*-invariant tensor fields on the Siegel upper halfplane can be viewed as vector valued Siegel modular forms with respect to this group

*ƒ¡*.

For our group the Satake compactification is the

*3*-dimensional projective space.

After observing the tensors on the Satake compactification the structure theorem(s) and Hilbert function(s) for the representation Sym2 become rather evident.

Here, we discovered a new strategy to retrieve structure theorems for other appropriate groups.

Examples executed by Freitag, Salvati Manni and partially the speaker include the groups of genus two

*ƒ¡*and

_{2}[4, 8]*ƒ¡*and even one of Igusafs subgroups of genus 3

_{2}[2, 4, 8]*ƒ¡*.

_{3}[2, 4]Using invariant theory we could reprove Aokifs structure theorem for

*ƒ¡*and Cleryfs, van der Geerfs and Grushevskyfs structure theorem for

_{2,0}[2]*ƒ¡*and Sym

_{2}[2]^{2}.

### 15:05--15:50 Tomoyoshi Ibukiyama (Osaka University)

Title :

Construction of liftings to vector valued Siegel modular forms

Abstract :

Using the Hayashida-Maass relation of Ikeda lift
and good differential operators, we construct
several liftings to vector valued Siegel modular
forms of integral or half-integral weight
from a pair of elliptic modular forms.

### 16:10--16:55 Norio Adachi (Waseda University)

Title :

'sacred' or 'profane' ?

Abstract :

What is mathematics? We present the following
tentative answer for discussion.

Mathematics takes a syntactical and a semantical
form; the syntactical form is usually called
pure mathematics, the semantical form applied
mathematics.

### 18:00--20:00 Banquet

## Thursday, 13th March

### 9:30--10:15 Mika Sakata (Kinki University)

Title :

Sum formula for mod *p* multiple zeta values

Abstract :

Poly-Bernoulli numbers were introduced and
studied by M.Kaneko as a generalization of
classical Bernoulli numbers.

He clarified the *p*-divisibility
of denominators of di-Bernoulli numbers.

On the other hand, poly-Bernoulli
numbers of negative index have combinatorial
interpretation.

In this talk, we plan to discuss their
*p*-divisibility and periodicity.

### 10:35--11:20 Minoru Hirose (Kyoto University)

Title :

On the theory of fans and its application to Shintani *L*-function
and Hecke *L*-function

Abstract :

Shintani *L*-function is a holomorphic
function of several variables defined by a
certain Dirichlet series.

A fan is a formal sum of cone regions.

I explain about the theory of fans and its
application to Shintani *L*-function
and Hecke *L*-function.

### 11:45--12:20 Fumitake Hyodo (Waseda University)

Title :

A formal power series of a Hecke ring associated with the Heisen-
berg Lie algebra

Abstract :

This talk studies a formal power series with
coefficients in a Hecke ring associated with the
Heisenberg Lie algebra.

We relate the series to the classical Hecke
series defined by E. Hecke, and prove that the
series has a property similar to the rationality
theorem of the classical Hecke series.

And then, our results recover the rationality
theorem of the classical Hecke series.