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Infinite Balanced Cover
Introduction.
To emphasize the following problem's significance,
we point out that,
for the 2020 complex quadratic fields K = Q(D^{1/2})
with discriminant 10^{6} < D < 0
and 3class group Cl_{3}(K) of type (3,3),
our
extensive computations of 2010 [Ma1]
and
our theories developed in 2009  2013 [Ma4]
have revealed that

the dominant part of exactly 936 fields has
a 3class tower of length l_{3}(K) = 2
(minimum D = 4027),

the second largest part of at least 411 fields has
a
3class tower of length l_{3}(K) = 3 [BuMa]
(minimum D = 9748),

the third largest part of 297 fields is
of the kind described by Theorem H.4GS,
which characterizes the
ground state of
capitulation type H.4 [Ma2]
(minimum D = 3896).
Theorem H.4GS.
(D. C. Mayer, 2009 [Ma3])
For a complex quadratic field K = Q(D^{1/2}), D < 0,
with 3class group Cl_{3}(K) of type (3,3)
the following conditions are equivalent:

The 3class groups of the four
unramified cyclic cubic extensions N_{1},…,N_{4} of K
are of the types Cl_{3}(N_{1}) ≅ (3,9)
and Cl_{3}(N_{i}) ≅ (3,3,3) for 2 ≤ i ≤ 4
(up to the selection of an ordering).

The Galois group of the second Hilbert 3class field F_{3}^{2}(K),
that is the maximal metabelian unramified 3extension of K,
is isomorphic to the unbalanced 3group SmallGroup(729,45)
of class 4, coclass 2, and derived length 2.
Problem.
(originally intended for the problem sessions of WCNT 2013 [Ma5])
By which arithmetical criteria
can we decide for a complex quadratic field
of the kind described by Theorem H.4GS
which of the infinitely many nonmetabelian
(balanced) Schur σgroups G_{n}, n ≥ 2,
given by
L. Bartholdi and M. R. Bush [BtBu]
(2007),
is isomorphic to the Galois group of the
maximal unramified pro3 extension F_{3}^{∞}(K) of K ?
Remark.
The group G_{n}, n ≥ 2, has second derived quotient
G_{n} / G_{n}^{''} ≅ SmallGroup(729,45)
[we express this fact by saying that the set { G_{n}  n ≥ 2 }
is the infinite balanced cover of SmallGroup(729,45)]
and is of order 3^{3n+2}, class 2n + 1, coclass n + 1,
and of unbounded derived length floor(log_{2}(3n+3)) for n → ∞.



Trade, Science, Art and Industry
Principal investigator of the
International Research Project
Towers of pClass Fields
over Algebraic Number Fields
supported by the Austrian Science Fund (FWF):
P 26008N25
