|On these pages, we present most recent results of our joint research, directly from the lab.|
K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237
A. Derhem, Capitulation dans les extensions quadratiques de corps de nombres cubiques cycliques, Thèse de doctorat, Université Laval, Québec, 1988
D. C. Mayer, Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), 831-847 and S55-S58
|Lattice Geometry of Cyclic Cubic Fields (2002/04/15)|
Having computed the regulators R of a large series of cyclic cubic fields L
with the aid of VORONOI's algorithm for signature space (3,0),
I was left with extensive information concerning the geometry of
the discrete MINKOWSKI images of their maximal orders O_L.
In fact, this was the reason why I had chosen the method of VORONOI
(firing cannon balls on sparrows)
instead of M.-N. GRAS' techniques to compute regulators.
The most striking feature of cyclic cubics as opposed to general totally real cubics
is the symmetry of the VORONOI chains and branches with respect to the three coordinate axes:
X-, Y-, and Z-axis have identical preperiod v, chain length pl,
and norms of lattice minima N(n^i(1)) (i = 0,...,v+pl)
(the associations from v to v+pl in these chains yield units uX, uY, uZ),
XY-, YZ-, and ZX-branch have identical length bl and unit association positions ap
(the associations from bl on the branches to ap on the chains yield further independent units uXY, uYZ, uZX),
and the same is true for the XZ-, YX-, and ZY-branches.
VORONOI's Main Theorem for signature space (3,0) states that a fundamental pair of units for L
is given by any one of the following: (uX,uXY), (uX,uXZ), (uY,uYZ), (uY,uYX), (uZ,uZX), (uZ,uZY).
A consequence of the symmetry is that a lattice is either of DIRICHLET type ,
where even (uX,uY), (uY,uZ), and (uZ,uX) are fundamental pairs of units,
or of Anti-DIRICHLET type (a special VORONOI type ),
where none of (uX,uY), (uY,uZ), and (uZ,uX) is a fundamental pair of units.
I. e., mixed VORONOI types cannot occur.
The singulets L1 with prime conductors f = q show different tendencies than
the doublets (L1,L2) with 2-prime conductors f = q1*q2,
the quadruplets (L1,L2,L3,L4) with 3-prime conductors f = q1*q2*q3, and
the octuplets (L1,...,L8) with 4-prime conductors f = q1*q2*q3*q4.
With increasing number of ramified primes dividing the conductor f, there is a trend
towards DIRICHLET type lattices, pure periodic chains (without preperiods), and smaller regulators.
However, with ascending conductors f, we have a tendency
towards VORONOI type lattices, mixed periodic chains (with preperiods), and bigger regulators.
1. In the first table (*) with all prime conductors f = q < 10^5,
I found the following statistics of Lattice Types,
HiChamps of preperiod v max, chain length pl max, and regulator R max, and
LoChamps of the regulator R min (we always have v min = 0 and pl min = 1),
with respect to conductors f in relative intervals of length 10k = 10000
(concerning the field counts compare my earlier communications):
2. In the second table (**) with all 2-prime conductors f = q1*q2 < 10^5:
3. In the third table (***) with all 3-prime conductors f = q1*q2*q3 < 10^5
with respect to conductors f in relative intervals of length 50k = 50000:
4. In the fourth table (****) with all 4-prime conductors f = q1*q2*q3*q4 < 10^5
with respect to conductors f in the absolute interval of length 100k = 100000:
It is important to point out that
such an extensive series of cyclic cubic fields,
where the conductors f are rapidly increasing,
is more adequate to show the typical geometric phenomena
with long periods, preperiods, chain lengths, and branch lengths
than a table of general totally real cubic fields,
where the conductors f and the lattice invariants remain modest,
the main feature is a broad coverage of quadratic field discriminants d,
and the increase of cubic field discriminants D = d*f^2 > 0
(a lot of them being fundamental, i. e., unramified with f = 1)
is mainly due to d and not to f.
The same effect arises for pure cubic fields with D = (-3)*f^2
as opposed to general complex cubic fields,
here on the side of negative discriminants D < 0.