| Dirichlet Type Lattices in Cyclic Cubic Fields (2002/04/26) |
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Dan (02/04/26):
Using VORONOI's algorithm for signature (3,0), published in his doctoral dissertation, we can determine the geometry of lattice minima in the discrete geometric image of O_L, the maximal order of any cyclic cubic field L, racing along the VORONOI HighWays in various directions. |
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Dan (02/04/26): These lattices have the nice property that
uXY = uZY = uY^-1, uYZ = uXZ = uZ^-1, uZX = uYX = uX^-1. This structure occurs with period length 1 for the conductors f = 7, 9, 13, 19, 37, 79, 97, 139, 163, 313, ..., with period length 3 for f = 43, 67, 151, 397, 541, ..., with period length 4 for f = 181, 211, 271, 421, ..., and with period length 6 for f = 127, 337, ... Chain in X-direction: no pre period v = 0, chain length pl_X = 1, N(n_X^1(1)) = 1 X-Branch in Y-direction: branch length bl_XY = 1, association on X-chain at position ap = 0, N(n_Y^1(1)) = N(n_X^0(1)) X-Branch in Z-direction: branch length bl_XZ = 1, association on X-chain at position ap = 0, N(n_Z^1(1)) = N(n_X^0(1)) The symmetry of lattices in cyclic cubic fields implies that the chains in Y-direction and in Z-direction have the same structure as the chain in X-direction. 
All DIRICHLET systems of units are fundamental: XY-system: index ( U : < uX,uY > ) = 1 YZ-system: index ( U : < uY,uZ > ) = 1 ZX-system: index ( U : < uZ,uX > ) = 1 which shows the symmetry again. Fundamental pairs of units are, of course, also given by: (uX,uXY),(uX,uXZ),(uY,uYZ),(uY,uYX),(uZ,uZX),(uZ,uZY). |