Joint research 2002

of Karim Belabas, Aïssa Derhem, and Daniel C. Mayer

Lattice Geometry of Cyclic Cubic Fields (2002/04/15)
Dan (02/04/10): 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.
Dan (02/04/19): 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.
Dan (02/04/10): 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): # Fields # DIRICHLET # VORONOI v max pl max R max R min f < 10k 612 157(25.7%) 455(74.3%) 419 654 3.4e4 5.3e-1 f < 20k 513 101(19.7%) 412(80.3%) 732 1654 6.4e4 2.1e1 f < 30k 486 76(15.6%) 410(84.4%) 658 1754 1.1e5 2.5e1 f < 40k 479 76(15.9%) 403(84.1%) 1351 1985 1.9e5 2.7e1 f < 50k 467 62(13.3%) 405(86.7%) 1398 1963 2.2e5 2.8e1 f < 60k 458 51(11.1%) 407(88.9%) 3149 2486 2.3e5 2.9e1 f < 70k 429 48(11.2%) 381(88.8%) 2918 3010 2.8e5 3.1e1 f < 80k 455 45(9.9%) 410(90.1%) 2382 3484 3.9e5 3.1e1 f < 90k 440 54(12.3%) 386(87.7%) 2419 4389 3.6e5 3.2e1 f < 100k 446 50(11.2%) 396(88.8%) 1969 3707 3.9e5 3.3e1 Total 4785 720(15.05%) 4065(84.95%) 3149 4389 3.9e5 5.3e-1 # DIRICHLET v = 0 v > 0 pl = 1 # VORONOI v = 0 v > 0 f < 10k 157 140(89.2%) 17 33 455 278(61.1%) 177 f < 20k 101 89(88%) 12 10 412 210(51.0%) 202 f < 30k 76 62(82%) 14 8 410 180(43.9%) 230 f < 40k 76 63(83%) 13 6 403 157(39.0%) 246 f < 50k 62 52(84%) 10 7 405 148(36.5%) 257 f < 60k 51 41(80%) 10 3 407 156(38.3%) 251 f < 70k 48 41(85%) 7 4 381 140(36.7%) 241 f < 80k 45 33(73%) 12 3 410 146(35.6%) 264 f < 90k 54 44(81%) 10 3 386 130(33.7%) 256 f < 100k 50 45(90%) 5 5 396 137(34.6%) 259 Total 720 610(84.7%) 110 82 4065 1682(41.38%) 2383
Dan (02/04/10): 2. In the second table (**) with all 2-prime conductors f = q1*q2 < 10^5: # Fields # DIRICHLET # VORONOI v max pl max R max R min f < 10k 788 333(42.3%) 455(57.7%) 147 419 9.3e3 1.3e1 f < 20k 796 224(28.1%) 572(71.9%) 262 667 2.3e4 2.2e1 f < 30k 786 208(26.5%) 578(73.5%) 590 1043 3.5e4 2.5e1 f < 40k 766 165(21.5%) 601(78.5%) 542 1214 5.2e4 2.7e1 f < 50k 770 168(21.8%) 602(78.2%) 645 1040 7.0e4 2.8e1 f < 60k 770 146(19.0%) 624(81.0%) 1276 1442 7.2e4 2.9e1 f < 70k 800 132(16.5%) 668(83.5%) 1369 1785 1.0e5 3.0e1 f < 80k 758 133(17.6%) 625(82.4%) 703 1575 1.3e5 3.1e1 f < 90k 772 129(16.7%) 643(83.3%) 1021 2707 1.3e5 3.2e1 f < 100k 720 131(18.2%) 589(81.8%) 1560 2048 1.3e5 3.3e1 Total 7726 1769(22.90%) 5957(77.10%) 1560 2707 1.3e5 1.3e1 # DIRICHLET v = 0 v > 0 pl = 1 # VORONOI v = 0 v > 0 f < 10k 333 313(94.0%) 20 82 455 352(77.4%) 103 f < 20k 224 209(93.3%) 15 22 572 358(62.6%) 214 f < 30k 208 195(93.8%) 13 18 578 362(62.6%) 216 f < 40k 165 149(90.3%) 16 15 601 350(58.2%) 251 f < 50k 168 148(88.1%) 20 17 602 321(53.3%) 281 f < 60k 146 133(91.1%) 13 15 624 336(53.8%) 288 f < 70k 132 114(86.4%) 18 10 668 356(53.3%) 312 f < 80k 133 119(89.5%) 14 9 625 304(48.6%) 321 f < 90k 129 115(89.2%) 14 6 643 311(48.4%) 332 f < 100k 131 119(90.8%) 12 6 589 290(49.2%) 299 Total 1769 1614(91.24%) 155 200 5957 3340(56.07%) 2617
Dan (02/04/18): 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: # Fields # DIRICHLET # VORONOI v max pl max R max R min f < 50k 1404 611(43.5%) 793(56.5%) 575 381 1.9e4 1.1e1 f < 100k 1728 484(28.01%) 1244(71.99%) 624 734 3.9e4 3.1e1 Total 3132 1095(34.96%) 2037(65.04%) 624 734 3.9e4 1.1e1 # DIRICHLET v = 0 v > 0 pl = 1 # VORONOI v = 0 v > 0 f < 50k 611 586(95.9%) 25 111 793 538(67.8%) 255 f < 100k 482 448(92.9%) 34 42 1241 809(65.2%) 432 Total 1095 1036(94.61%) 59 153 2037 1350(66.27%) 687
Dan (02/04/19): 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: # Fields # DIRICHLET # VORONOI v max pl max R max R min f < 100k 208 117(56.2%) 91(43.8%) 121 253 8.1e3 2.9e1 # DIRICHLET v = 0 v > 0 pl = 1 # VORONOI v = 0 v > 0 f < 100k 117 114(97.4%) 3 27 91 70(77%) 21
Dan (02/04/15): 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.