Abstract
Let X ⊂ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] be a possibly singular hypersurface of degree d ≤ n , 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 H i ([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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