The 18th Number Theory Meeting at Waseda Univ.
Tuesday, 11th March, 2014 -- Thursday, 13th March, 2014
Keiichi Komatsu (Waseda University),
Kiichiro Hashimoto (Waseda University),
Manabu Ozaki (Waseda University), Hiroshi Sakata (Waseda University Senior High School)
JSPS Grant-in-Aid for Scientic Research (C) (24540030,Keiichi Komatsu) and (21540030, Manabu Ozaki).
Tuesday, 11th March
10:00--10:45 Shuichi Ohtake (Waseda University)
Orthogonal decompositions of integral trace forms of cyclotomic fields
Trace form is the symmetric F-bilinear form on K:= F[x]=(f(x)) defined by (x,y) ¨ traceK/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 OK of the number field K defines a symmetric bilinear form over the ring of rational integers Z on OK - called the integral trace form of 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)
On Greenberg's conjecture for CM-fields
Let k/Q be a finite extension and p an odd prime number.
Let K/k be the maximal multiple Zp-extension, and let Gal(K/k) [¯^] Zpd.
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 © with coefficients in Zp of d-variables.
Then Greenberg conjectured that X is pseudo-null over ©.
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) Ép(k+) = Êp(k+) = Ëp(k+) = 0, where É, Ê and Ë denote Iwasawa invariants and k+ denotes the maximal totally real subfield of k.
14:00 -- 14:45 Manabu Ozaki (Waseda University)
The Neukirch-Uchida theorem for a certain class of number fields of infinite degree
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)
Reconstruction of a Number Field From the Absolute Galois Group
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)
A control theorem for the torsion Selmer pointed set
Selmer groups are important arithmetic invariants of Galois representations.
Minhyong Kim defined the Selmer variety which is a
non-abelian analogue of the Qp-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)
Local Ã-isomorphisms for rank two p-adic representations of Gal((Q_p)^[/Q_p) and a functional equation of Kato's Euler system
Local Ã-isomorphisms are conjectural bases of the determinants of the Galois cohomologies of p-adic representations of Gal(Qp[/Qp) which p-adically interpolate local constants (Ã-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 GL2(Qp), we define such bases for (almost) all rank two torsion p-adic representations.
We show that our Ã-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 Ã-conjecture.
Wednesday, 12th March
10:00--10:45 Masao Tsuduki (Sophia University)
Equidistribution and subconvexity bound related to certain L-values
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)
Determination of the structure of vector valued Siegel modular forms by using Jacobi forms
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
Geometrically proven structure theorems for vector valued Siegel modular forms
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 Sym2 and Igusafs subgroup ¡2[2, 4]. They rest on the well known fact that ¡-invariant tensor fields on the Siegel upper halfplane can be viewed as vector valued Siegel modular forms with respect to this group ¡.
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 ¡2[4, 8] and ¡2[2, 4, 8] and even one of Igusafs subgroups of genus 3 ¡3[2, 4].
Using invariant theory we could reprove Aokifs structure theorem for ¡2,0 and Cleryfs, van der Geerfs and Grushevskyfs structure theorem for ¡2 and Sym2.
15:05--15:50 Tomoyoshi Ibukiyama (Osaka University)
Construction of liftings to vector valued Siegel modular forms
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)
'sacred' or 'profane' ?
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.
Thursday, 13th March
9:30--10:15 Mika Sakata (Kinki University)
Sum formula for mod p multiple zeta values
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)
On the theory of fans and its application to Shintani L-function and Hecke L-function
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)
A formal power series of a Hecke ring associated with the Heisen- berg Lie algebra
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.