IPAD and IPOD 2016



IPAD and IPOD

Index-p Abelianization Data

Index-p Obstruction Data


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



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.

  1. 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.
  2. 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 .


International Research Project


Daniel C. Mayer
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:





Open Problems

In 2016, we are going to apply the nearly complete theory of p-class tower groups,
which was developed in the Tetralogy and in the Trilogy mentioned above,
to extensive series of number fields with p-class groups of the types (p,p), (p2,p) and (p,p,p).
Since it is a hard problem already to compute the IPOD of a number field,
it will be a challenge to start using iterated IPODs of second order
by extending the computational techniques to prime power degree extensions.

The first success in this direction was published at the beginning of 2016 in

[12] Index-p abelianization data of p-class tower groups, Part II, 2016 .

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