Our targets for 2007 / 2008

Here we further pursue our well-known intricate targets:

To determine the capitulation type of the 2020 complex quadratic fields K
with discriminant -10⁶ < d < 0 and 3-class group of type (3,3).
These objects were first listed in Supercomputing Centre 2006.
To determine the capitulation type of the 149 real quadratic fields K
with discriminant 0 < d < 10⁶ and 3-class group of type (3,3).
These objects were first listed in Supercomputing Centre 2006.

Together with the 3-class numbers of the associated quadruplets (L₁,…,L₄) of cubic fields,
the capitulation type gives the isomorphism class of the 2-stage metabelian 3-group
G=Gal(K₂|K) in CBF(m,n) of automorphisms of the 2^nd Hilbert 3-class field K₂ over K,
with the aid of our most recent sophisticated theorems in algebraic number theory [1].

Remarks added in 2010:

The first target has been reached on the 15^th of July, 2010,
with the completion of the complex quadratic capitulation project.
The second target has been reached on the 17^th of December, 2009,
with the completion of the real quadratic capitulation project,
and has been extended to the 2576 real quadratic fields K
with discriminant 0 < d < 10⁷ and 3-class group of type (3,3)
on the 20^th of March, 2010.

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References:

[1] Daniel C. Mayer,
Two-Stage Towers of 3-Class Fields over Quadratic Fields,
(Latest Update)
Univ. Graz, 2008.

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