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 $\ell$-adic cohomology to treat extensions of odd degree as well.
- Publication:
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arXiv e-prints
- Pub Date:
- November 1997
- DOI:
- 10.48550/arXiv.alg-geom/9711021
- arXiv:
- arXiv:alg-geom/9711021
- Bibcode:
- 1997alg.geom.11021L
- Keywords:
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- Algebraic Geometry
- E-Print:
- 44 pages, Plain TeX