Signed fundamental domains for totally real number fields

Abstract

We give a signed fundamental domain for the action on Rn+ of the totally positive units E+ of a totally real number field k of degree n. The domain {(Co, wo)}o is signed since the net number of its intersections with any E+-orbit is 1, that is, for any x [épsilon]Rn+, [sigma]oESn-1 [sigma]eEE+ WoXco (EX)= 1

Here, XCo is the characteristic function of Co, Wo = |1 is a natural orientation of the n-dimensional k-rational cone Co CRn+, and the inner sum is actually finite. Signed fundamental domains are as useful as Shintani fs true ones for the purpose of calculating abelian L-functions. They have the advantage of being easily constructed from any set of fundamental units, whereas in practice there is no algorithm producing Shintani fs k-rational cones. Our proof uses algebraic topology on the quotient manifold Rn+/E+. The invariance of the topological degree under homotopy allows us to control the deformation of a crooked fundamental domain into nice straight cones. Crossings may occur during the homotopy, leading to the need to subtract some cones.