International Conferences 2016
July 25 - 27, 2016:
2nd International Conference
on Groups and Algebras
ICGA 2016, Suzhou
Presentation:
p-Capitulation over Number Fields
with p-Class Rank Two,
Suzhou,
China
November 11 - 12, 2016:
International Colloquium of Algebra,
Number Theory, Cryptography,
and Information Security
ANCI 2016, Taza
Invited Lecture:
Recent Progress in Determining
p-Class Field Towers,
Taza,
Morocco
Our services to the mathematical community:
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Definitions
Let G be a group with a subgroup H < G of finite index (G:H) = n.
Suppose that a left transversal of H in G is given by
G = g(1)H + … + g(n)H.
Then the left action of a group element x in G causes a permutation of the left cosets
xg(i)H = g(s(i))H, for i in {1,…,n},
with a permutation s in the symmetric group S(n) of degree n.
The Artin transfer T(G,H): G → H/H' from G to the abelianization H/H' of H
is a group homomorphism defined by
T(G,H)(x) = g(s(1))-1xg(1)* … * g(s(n))-1xg(n) * H'.
In particular, if p is a prime number and G is a finite p-group or an infinite pro-p group
then we denote by Lyr(1,G) the first layer of normal subgroups H < G with index (G:H) = p in G.
If G has a finite abelianization G/G' then Lyr(1,G) is a finite set
and we call
the family of transfer targets (abelianizations)
t(G) = ( H/H' ) with H in Lyr(1,G)
the Index-p Abelianization Data, briefly the IPAD of G, and
the family of transfer kernels
k(G) = ( Ker(T(G,H) ) with H in Lyr(1,G)
the Index-p Obstruction Data, briefly the IPOD of G.
The pair A(1,G) = ( t(G),k(G) ) is the first layer of the Artin pattern A(G) of G.
Examples
Towers of Type H.4
Towers of Type G.19
Background and Aims
The Artin pattern A(G) of a pro-p group G has turned out to be the decisive information for solving
the problem of the Hilbert p-class field tower F(p,∞,K) of an algebraic number field K.
Within the frame of our
International Scientific Research Project
with title
Towers of p-class fields over algebraic number fields,
which is supported financially by the Austrian Science Fund (FWF):
P 26008-N25
,
this goal of identifying the p-class tower F(p,∞,K) has been approached in several steps.
-
The metabelianization G/G'' of the Galois group G of the p-class tower,
which describes the lowest two stages of the tower,
K < F(p,1,K) < F(p,2,K),
was investigated in four subsequent articles, forming a Tetralogy,
[1] The second p-class group of a number field, 2012
,
[2] Transfers of metabelian p-groups, 2012
,
[3] Principalization algorithm via class group structure, 2014
,
[4] The distribution of second p-class groups on coclass graphs, 2013
.
In [3] and [4], the IPAD was called the transfer target type, briefly TTT,
and the IPOD was called the transfer kernel type, briefly TKT.
-
Inspired by our cooperation with Michael R. Bush and the resulting joint paper
3-class field towers of exact length 3, 2015
,
we pushed forward beyond the two-stage towers into the strange realm of
non-metabelian p-groups with derived length 3,
K < F(p,1,K) < F(p,2,K) < F(p,3,K),
whose investigation requires
iterated IPADs of second order developed systematically in a Trilogy,
[5] Periodic bifurcations in descendant trees of finite p-groups, 2015
,
[6] Index-p abelianization data of p-class tower groups, 2015
,
[7] Artin transfer patterns on descendant trees of finite p-groups, 2016
,
with parallel applications to special types of algebraic number fields in the following articles,
[8] Periodic sequences of p-class tower groups, 2015
,
[9] New number fields with known p-class tower, 2016
,
and in cooperation with Abdelmalek Azizi, Abdelkader Zekhnini and Mohammed Taous,
[10] Principalization of 2-class groups of type (2,2,2) of biquadratic fields, 2015
,
respectively with Abdelmalek Azizi, Mohamed Talbi, Mohammed Talbi, and Aissa Derhem,
[11] The group Gal( K32 / K ) for K = Q( (-3)1/2, D1/2 ) of type (3,3), 2016
.
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International Research Project
Principal investigator of the
International Research Project
Towers of p-Class Fields
over Algebraic Number Fields
supported by the Austrian Science Fund (FWF):
P26008-N25
Time Schedule:
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