A geometric approach to the fundamental lemma for unitary groups

Details A geometric approach to the fundamental lemma for unitary groups A geometric approach to the fundamental lemma for unitary groups

1997-11-19

arxiv.org/abs/alg-geom/9711021v1

Mathematics

Algebraic Geometry

44 pages, Plain TeX

Scientific paper

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 $\ell$-adic cohomology to treat extensions of odd degree as well.

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