A geometric approach to the fundamental lemma for unitary groups

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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 kf-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.