| Voronoi 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): First an example of inverse associations,
where uYX = uXY^-1, uZY = uYZ^-1, uXZ = uZX^-1. This structure occurs for the conductors f = 61, 331, 547, ... Chain in X-direction: no pre period v = 0, chain length pl_X = 6, N(n_X^6(1)) = 1 X-Branch in Y-direction: branch length bl_XY = 1, association on X-chain at position ap = 2, N(n_Y^1(1)) = N(n_X^2(1)) X-Branch in Z-direction: branch length bl_XZ = 2, association on X-chain at position ap = 1, N(n_Z^2(1)) = N(n_X^1(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. 
None of the DIRICHLET systems of units is fundamental: XY-system: index ( U : < uX,uY > ) = 3 YZ-system: index ( U : < uY,uZ > ) = 3 ZX-system: index ( U : < uZ,uX > ) = 3 which shows the symmetry again. Fundamental pairs of units are given by: (uX,uXY),(uX,uXZ),(uY,uYZ),(uY,uYX),(uZ,uZX),(uZ,uZY). |
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Dan (02/04/26): Next an example without inverse associations:
this structure occurs for the conductors f = 31, 103, 277, ... Chain in X-direction: no pre period v = 0, chain length pl_X = 8, N(n_X^8(1)) = 1 X-Branch in Y-direction: branch length bl_XY = 1, association on X-chain at position ap = 4, N(n_Y^1(1)) = N(n_X^4(1)) X-Branch in Z-direction: branch length bl_XZ = 1, association on X-chain at position ap = 2, N(n_Z^1(1)) = N(n_X^2(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. 
None of the DIRICHLET systems of units is fundamental: XY-system: index ( U : < uX,uY > ) = 7 YZ-system: index ( U : < uY,uZ > ) = 7 ZX-system: index ( U : < uZ,uX > ) = 7 which shows the symmetry again. Fundamental pairs of units are given by: (uX,uXY),(uX,uXZ),(uY,uYZ),(uY,uYX),(uZ,uZX),(uZ,uZY). |