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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(D1/2)
with discriminant -106 < D < 0
and 3-class group Cl3(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 3-class tower of length l3(K) = 2
(minimum |D| = 4027),
-
the second largest part of at least 411 fields has
a
3-class tower of length l3(K) = 3 [BuMa]
(minimum |D| = 9748),
-
the third largest part of 297 fields is
of the kind described by Theorem H.4-GS,
which characterizes the
ground state of
capitulation type H.4 [Ma2]
(minimum |D| = 3896).
Theorem H.4-GS.
(D. C. Mayer, 2009 [Ma3])
For a complex quadratic field K = Q(D1/2), D < 0,
with 3-class group Cl3(K) of type (3,3)
the following conditions are equivalent:
-
The 3-class groups of the four
unramified cyclic cubic extensions N1,…,N4 of K
are of the types Cl3(N1) ≅ (3,9)
and Cl3(Ni) ≅ (3,3,3) for 2 ≤ i ≤ 4
(up to the selection of an ordering).
-
The Galois group of the second Hilbert 3-class field F32(K),
that is the maximal metabelian unramified 3-extension of K,
is isomorphic to the unbalanced 3-group 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.4-GS
which of the infinitely many non-metabelian
(balanced) Schur σ-groups Gn, n ≥ 2,
given by
L. Bartholdi and M. R. Bush [BtBu]
(2007),
is isomorphic to the Galois group of the
maximal unramified pro-3 extension F3∞(K) of K ?
Remark.
The group Gn, n ≥ 2, has second derived quotient
Gn / Gn'' ≅ SmallGroup(729,45)
[we express this fact by saying that the set { Gn | n ≥ 2 }
is the infinite balanced cover of SmallGroup(729,45)]
and is of order 33n+2, class 2n + 1, coclass n + 1,
and of unbounded derived length floor(log2(3n+3)) for n → ∞.
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Trade, Science, Art and Industry
Principal investigator of the
International Research Project
Towers of p-Class Fields
over Algebraic Number Fields
supported by the Austrian Science Fund (FWF):
P 26008-N25
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