Abstract

Let X ⊂ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] be a possibly singular hypersurface of degree dn, defined over a finite field [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]. We show that the diagonal, suitably interpreted, is decomposable. This gives a proof that the eigenvalues of the Frobenius action on its ℓ-adic cohomology Hi([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]), for ℓ ≠ char([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]), are divisible by q, without using the result on the existence of rational points by Ax and Katz.

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Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 193-207
Launched on MUSE
2005-02-02
Open Access
No
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