A geometric approach to the fundamental lemma for unitary groups
Abstract
We consider from a geometric point of view the conjectural fundamental lemma of Langlands and Shelstad for unitary groups over a local field of positive characteristic. We introduce projective algebraic varieties over the finite residue field
k
and interpret the conjecture in this case as a remarkable identity between the number of
k
-rational points of them. We prove the corresponding identity for the numbers of
k
f
-rational points, for any extension of even degree
f
of
k
. The proof uses the local intersection theory on a regular surface and Deligne's theory of intersection multiplicities with weights. We also discuss a possible descent argument that uses
ℓ
-adic cohomology to treat extensions of odd degree as well.