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1. To advance European science
to the forefront
of international research
and to stabilize this position.
2. To strengthen cooperation
with international research centers:

HarishChandra Research Institute,
Allahabad, India

Washington and Lee University,
Lexington, Virginia, USA

Université Mohammed Premier,
Oujda, Morocco

Australian National University,
Canberra, Capital Territory

Aichi University of Education,
Nagoya, Japan

University of Calgary,
Alberta, Canada
International Conferences:
July 1  5, 2019:
31st Journées Arithmétiques
JA 2019, Istanbul, Turkey

Progressive Innovations and
Outstanding Scientific Achievements:
Unexpected Discoveries on 26 April 2018

Although it seems to be an arithmetic invariant,
the degree [F_{p}^{∞}(K):K] of the
maximal unramified prop extension F_{p}^{∞}(K)
over an algebraic number field K
with assigned Artin transfer pattern AP(K)=(τ(K),κ(K))
cannot be determined by number theoretic methods in general.
The group theoretic strategy of pattern recognition must be employed
on a bounded root region of a suitable descendant tree of finite nonabelian pgroups.
During the CSASC Conference in 2011 at the Danube University Krems,
I presented the Principalization Algorithm via Class Group Structure
for quadratic number fields K=Q(d^{1/2}) with non elementary 3class group Cl_{3}(K)∼(21),
which is based on the fact that, for certain Artin patterns,
the components τ(K) and κ(K) determine each other uniquely
(τ(K) denotes the abelian quotient invariants (AQI) of first order
and κ(K) denotes the punctured transfer kernel type (pTKT) of K).
This correspondence is expressed by the following Theorem:
τ(K)∼((31)^{2},(21^{2})^{2}) if and only if κ(K)∼(124;1), called type D.11,
τ(K)∼((31)^{3},21^{2}) if and only if κ(K)∼(123;4), called type E.12,
τ(K)∼((31)^{3},1^{4}) if and only if κ(K)∼(111;4), called type B.7.
Theoretical results were underpinned by explicit computation of all Artin patterns
for discriminants in the range 10^{6} < d < 10^{7}.

In 2015, Michael R. Bush extended my range of discriminants considerably to 10^{8} < d < 10^{9}
classifying by the first component τ(K) and abstaining from the fine structure splitting by means of κ(K).
A comparison of the above mentioned three pTKTs with respect to the distribution of their absolute frequencies
in four partial ranges of discriminants reveals a very dense population for imaginary quadratic fields:
pTKT

10^{5} < d < 0

10^{6} < d < 0

10^{7} < d < 0

10^{8} < d < 0

10^{8} < d < 0

D.11

14

406

4974

55310

45.2%

E.12

13

75

844

9335

7.6%

B.7

3

64

799

9000

7.4%

total

48

875

10811

122444

100%


However, for real quadratic fields, the population turns out to be rather sparse:
pTKT

0 < d < 10^{6}

0 < d < 10^{7}

0 < d < 10^{8}

0 < d < 10^{9}

0 < d < 10^{9}

D.11

1

16

221

2844

4.8%

E.12

0

3

34

517

0.9%

B.7

0

2

35

489

0.8%

total

12

271

4679

59081

100%


For the punctured transfer kernel type D.11,
the two possible metabelian 3groups Gal(F_{3}^{2}(K)/K) ∼ <729,i> with i∈{14,15}
are Schur σgroups, whence F_{3}^{∞}(K)=F_{3}^{2}(K) is of degree 729 over K.
Theorem: The following three conditions are equivalent:
(1) τ(K)∼((31)^{2},(21^{2})^{2}),
(2) κ(K)∼(124;1), called type D.11,
(3) Gal(F_{3}^{∞}(K)/K) ∼ <729,i> with i∈{14,15}.

Now I come to the unexpected discoveries:
Artin patterns of second order admit deeper insight about type E.12.
Theorem: Let K=Q(d^{1/2}) be a quadratic field with
3class group Cl_{3}(K)∼(21) and abelian quotient invariants of first order τ(K)∼((31)^{3},21^{2})
(which is equivalent to a punctured transfer kernel type κ(K)∼(123;4), called type E.12).
Then abelian quotient invariants of second order τ^{(2)}(K) can be used for the following criteria:
(1) If Gal(F_{3}^{∞}(K)/K) ∼ <2187,178>#2;2 of order 19683, then
Gal(F_{3}^{2}(K)/K) ∼ <6561,1734> and
τ^{(2)}(K)∼
(21;[(31);2^{2}1,(41)^{3}]^{3},[21^{2};2^{2}1,(31^{2})^{6},(32)^{6}]).
(2) If Gal(F_{3}^{∞}(K)/K) ∼ <2187,188>#2;2 of order 19683, then
Gal(F_{3}^{2}(K)/K) ∼ <6561,1772> and
τ^{(2)}(K)∼
(21;[(31);2^{2}1,(41)^{3}]^{3},[21^{2};2^{2}1,(32)^{12}]).
(3) τ^{(2)}(K)∼
(21;[(31);2^{2}1,(41)^{3}],[(31);2^{2}1,(31)^{3}]^{2},[21^{2};2^{2}1,(21^{2})^{6},(2^{2})^{6}])
if and only if K is real with d > 0 and
Gal(F_{3}^{∞}(K)/K) ∼ <6561,1735>, Gal(F_{3}^{2}(K)/K) ∼ <2187,178>.
(4) τ^{(2)}(K)∼
(21;[(31);2^{2}1,(41)^{3}],[(31);2^{2}1,(31)^{3}]^{2},[21^{2};2^{2}1,(2^{2})^{12}])
if and only if K is real with d > 0 and
Gal(F_{3}^{∞}(K)/K) ∼ <6561,1773>, Gal(F_{3}^{2}(K)/K) ∼ <2187,188>.
Remark: The criteria in item (3) and (4) of the theorem are necessary and sufficient.
However, the inverse implications in item (1) and (2) of the theorem cannot be proved rigorously,
since counterexamples can occur (observe that, for i∈{178,188}, the Schur σgroups <2187,i>#2;2 and
<2187,i>#2;1#1;2#2;2 share common AQI of second order τ^{(2)}(K)
but the latter has a different metabelianization <6561,j>#1;2 with j∈{1733,1771}).
Nevertheless the inverse implications are true with high probability, in a statistical sense.

Examples: The abelian quotient invariants of second order τ^{(2)}(K)∼
(1) (21;[(31);2^{2}1,(41)^{3}]^{3},[21^{2};2^{2}1,(31^{2})^{6},(32)^{6}])
occur for d∈{19427,42591,49576},
where Gal(F_{3}^{∞}(K)/K) must be a Schur σdescendant of <2187,178>,
(2) (21;[(31);2^{2}1,(41)^{3}]^{3},[21^{2};2^{2}1,(32)^{12}])
occur for d∈{5703,19919,27635,57336,61771},
where Gal(F_{3}^{∞}(K)/K) must be a Schur σdescendant of <2187,188>,
(3) (21;[(31);2^{2}1,(41)^{3}],[(31);2^{2}1,(31)^{3}]^{2},[21^{2};2^{2}1,(21^{2})^{6},(2^{2})^{6}])
do not occur for 0 < d < 10^{7},
(4) (21;[(31);2^{2}1,(41)^{3}],[(31);2^{2}1,(31)^{3}]^{2},[21^{2};2^{2}1,(2^{2})^{12}])
occur for d∈{1893032,9682121,9698952},
where Gal(F_{3}^{∞}(K)/K) ∼ <6561,1773> is determined uniquely.
(These three are the only real quadratic fields of type E.12 with discriminant 0 < d < 10^{7}.)
Preprints of peer reviewed publications:
Isoclinic propagation of algebraic invariants
Coperiodicity isomorphisms between forests of finite pgroups
Successive approximation of pclass towers
Deep transfers of pclass tower groups
Modeling rooted intrees by finite pgroups
Recent progress in determining pclass field towers


Principal Investigator and
Project Leader of several
International Scientific Research Lines:
