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Units and Satake's conjecture

The next collection of pictures originated in a project with R. Sczech and J. Sturm. Let F be a totally real number field, and we again consider the action of the totally positive units on the totally positive chamber Y. There is an F-rational lattice M in the Minkowski space X, and one takes a finite-index subgoup V of the totally positive units that preserves M. In the Shintani setup M was the ring of integers O_F, but now we consider a general F-lattice (M doesn't even have to be an order). Now just as in the Shintani setup we consider an infinite fan S of F-rational simplicial cones that fills out Y and has a V action.

Associated to the pair (M,V) are two objects, an L function and a cusp singularity. Satake's conjecture says that special values of the L function can be computed geometrically using intersection numbers on a certain smooth variety associated to the cusp singularity. This conjecture generalizes Hirzebruch's well-known theorem giving an arithmetic interpretation of the "signature defect" of Hilbert modular surface cusps.

Sczech, Sturm, and I proved this conjecture by completing a program initiated by Sczech. An essential step in the proof involved explicitly estimating an infinite sum of rational functions over the fan, and showing that it converged to a certain value. In order to do this, we needed to construct another F-rational fan S', again with a V action, with the following key property: there exists a sequence of convex F-rational cones C_1, C_2, ... that exhaust Y and such that each C_i is a union of cones of S'.

We proved that such a fan S' exists by explicitly constructing it, which involved carefully choosing a finite-index subgroup of V. Even after we had all the essential ideas and the proof, I felt like it was necessary to take a look. So I made the following pictures. Again the basic technology is the same: gp, perl, and geomview.

In this case the field is quartic; to draw pictures, we projectivize X=R^4 and look in P^3 = P^3(R). There are 16 orthants in X, and these projectivize to 8 tetrahedra in P^3. Any polyhedral cone projectivizes to a polytope in P^3, so we can get a feel for what's going on by looking at these polytopes. In the pictures, we draw the main tetrahedron corresponding to the totally positive chamber.

Our construction goes like this. We take a set of 4 infinite units {u_1, u_2, u_3, u_4}, any 3 of which have nonzero regulator. The set is chosen so that each of the four "continuous unit sheets" given by all monomials of the form {u_i^a u_j^b u_k^c | a+b+c=0, a,b,c real} has certain properties (basically we want each sheet to avoid one of the vertices of the tetrahedron as a,b,c get large). Then we can form the polyhedra C_i as the convex hull of the vertices {u_i^a u_j^b u_k^c | a+b+c=n_i, a,b,c integral, suitable n_i}. These can be represented as a "stack of small tetrahedra", and these small tetrahedra become the cones in S'.

That's probably too confusing to follow ... the reader will have to refer to the paper.

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Revised: Fri Oct 1 22:49:08 EDT 2010
Paul Gunnells
gunnells at math dot umass dot edu