Fame For Styria 2014: Infinite Balanced Cover

Challenges arising from an

Infinite Balanced Cover

Town Hall Graz, at night
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Infinite Balanced Cover

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
  1. the dominant part of exactly 936 fields has
    a 3-class tower of length l3(K) = 2
    (minimum |D| = 4027),
  2. the second largest part of at least 411 fields has
    a 3-class tower of length l3(K) = 3 [BuMa]
    (minimum |D| = 9748),
  3. 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:
  1. 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).
  2. 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 ?

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 → ∞.

Town Hall Graz, Figures
Trade, Science, Art and Industry
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):
P 26008-N25

Bibliographical References:

[BtBu] L. Bartholdi and M. R. Bush,
Maximal unramified 3-extensions of imaginary quadratic fields and SL2Z3,
J. Number Theory 124 (2007), 159 - 166.

[BuMa] M. R. Bush and D. C. Mayer,
3-class field towers of exact length 3,
J. Number Theory, 147 (2015), 766 - 777, DOI 10.1016/j.jnt.2014.08.010.

[Ma1] D. C. Mayer,
The second p-class group of a number field,
Int. J. Number Theory 8 (2012), no. 2, 471 - 505, DOI 10.1142/S179304211250025X.

[Ma2] D. C. Mayer,
Transfers of metabelian p-groups,
Monatsh. Math. 166 (2012), no. 3 - 4, 467 - 495, DOI 10.1007/s00605-010-0277-x.

[Ma3] D. C. Mayer,
Principalization algorithm via class group structure
J. Théor. Nombres Bordeaux, 26 (2014), no. 2, 415 - 464.

[Ma4] D. C. Mayer,
The distribution of second p-class groups on coclass graphs,
J. Théor. Nombres Bordeaux 25 (2013), no. 2, 401 - 456, DOI 10.5802/jtnb.842.
27th Journées Arithmétiques (JA) 2011,
Faculty of Mathematics and Informatics, Vilnius University,
Vilnius, Lithuania, 2011.

[Ma5] D. C. Mayer,
Class towers and capitulation over quadratic fields ,
West Coast Number Theory (WCNT) 2013,
Asilomar Conference Grounds, Pacific Grove,
Monterey, California, 2013.

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