The first series of pictures involves the following setup. Let F be a number field of degree n, and let X be its "Minkowski space" : this is real vector space given by tensoring F with the real numbers R. The space X is isomorphic to R^r \times C^s, where r+2s=n, and people who like the geometry of numbers like to study X and its relation to F.

Now T. Shintani proved a beautiful theorem about X and F, as first explained to me by R. Sczech. Let O_F be the ring of integers of F, and let U be the subgroup of units. In X, the ring O_F becomes a lattice L, and U acts on X and preserves L. One can ask what the U action looks like, and Shintani showed that there is a fundamental domain C for U consisting of a finite number of open L-rational polyhedral cones.

The first two series of pictures try to take a look at Shintani's decomposition when F is real cubic or complex cubic.

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Revised: Fri Oct 1 22:49:08 EDT 2010

Paul Gunnells

gunnells at math dot umass dot edu