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Let X be a smooth projective variety over a finite field $\mathbb{F}_{q}$ of caracteristic p and let l be a prime different from p. We consider the following statement :

(A) The action of the Frobenius on the etale cohomology $H^{i}_{et}( X_{\overline{F}_{q}}, \mathbb{Q}_{l})$ is semisimple.

How to suppress the projective hypothesis is the subject of the mathoverflow question link text

(A) is true in the following cases :

1) X an abelian variety (and so for X a curve via the jacobian). As mentionned in comment by Emerton, it is a consequence of the Weil's work on the Riemann hypothesis in this case. Fix a polarization on A. For x an endomorphism of X which gives an endomorphim on $H^{1}_{et}$, we can define an endomorphism x' (' : "Rosati involution") by $x' = *x^{T}*$ where in the middle we have the transposition with respect to the intersection product and * comes from the duality theory of abelian varieties ( the polarisation gives a identification betwenn between $H^{1}_{et}(A)$ et$H^{1}_{et}(\check{A})$). Weil proved that Tr(xx')>0 if x is non-zero. Let F be the (geometric) Frobenius. For $x = q^{-1/2}F$, we have x'=x. x'=$x^{-1}$. So Tr(aa') is a definite positive bilinear form on the $\mathbb{Q}$ \mathbb{Q}$ algebra generated by x and is preserved by multiplication by x : multiplication by x is so unitary which shows that x is semi-simple (and eigenvalues of modulus one gives the Riemann hypothesis). 2) X a K3 surface. As mentionned in comment by shenghao, it is a consequence of the work of Deligne : link text The result is deduced from the case of abelian varieties via the Kuga-Satake construction (of course there is a non-trivial thing to do because Kuga-Satake construction is a priori of transcendental nature but Deligne did it). For X general, (A) is conjectured. It is a consequence of standard conjectures. More precisely, things should work as in the case of abelian varieties. We can still define x -> x' at the cohomological level but Tr(xx')>0 is conjectural : standard conjecture of Lefschetz type imply x' algebraic if x is which permits to use a trace formula expressing Tr(xx') as an intersection product. The positivity should then be a consequence of standard conjecture of Hodge type. For more details, as mentionned in comment by Damian Rössler, see Kleiman "The standard conjectures" (whose some details depend on Kleiman, "Algebraic cycles and the Weil conjectures"). 2 added 2354 characters in body; deleted 3 characters in body; deleted 1 characters in body; edited body The semi-simplicity Let X be a smooth projective variety over a finite field $\mathbb{F}_{q}$ of caracteristic p and let l be a prime different from p. We consider the following statement : (A) The action of the frobenius Frobenius on the l-adic etale cohomology of a $H^{i}_{et}( X_{\overline{F}_{q}}, \mathbb{Q}_{l})$ is semisimple. How to suppress the projective smooth variety over a finite field hypothesis is the subject of the mathoverflow question link text (A) is true in the following cases : 1) X an abelian variety (and so for X a conjecturecurve via the jacobian).It As mentionned in comment by Emerton, it is a consequence of the standard conjecturesWeil'swork on the Riemann hypothesis in this case.It has beeen proved Fix a polarization on A. For x an endomorphism of X which gives an endomorphim on $H^{1}_{et}$, we can define an endomorphism x' (' : "Rosati involution") by $x' = *x^{T}* $where in some cases the middle we have the transposition with respect to the intersection productand * comes from the duality theory of abelian varieties ( the polarisation gives a identification betwenn $H^{1}_{et}(A)$ et$H^{1}_{et}(\check{A})$).Weil proved that Tr(xx')>0 if x is non-zero. Let F be the (geometric) Frobenius. For$x = q^{-1/2}F$, we have x'=x. So Tr(aa') is a definite positive bilinear form on the$\mathbb{Q}\$ algebra generated by x and is preserved by multiplication by x :it multiplication by x is true for curvesso unitary which shows that x is semi-simple (and eigenvalues of modulus one gives the Riemann hypothesis).

2) X a K3 surface. As mentionned in comment by shenghao, this case it is a consequence of all the work of Faltings on Mordell Deligne : link textThe result is deduced from the case of abelian varieties via the Kuga-Satake construction (of course there is a non-trivial thing to do becauseKuga-Satake construction is a priori of transcendental nature but Deligne did it).

For X general, (A) is conjectured. It is a consequence of standard conjectures. More precisely, things should work as in the case of abelian varieties. We can still define x -> x' at the cohomological level but Tr(xx')>0 is conjectural : standard conjecture of Lefschetz type imply x' algebraic if x is which permits to use a trace formula expressing Tr(xx') as an intersection product. The positivity should then be a consequence of standard conjecture of Hodge type.For more details, as mentionned in comment by Damian Rössler, see Kleiman "The standard conjectures" (whose some details depend on Kleiman, "Algebraic cycles and the Weil conjectures").

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The semi-simplicity of the action of the frobenius on the l-adic cohomology of a projective smooth variety over a finite field is a conjecture. It is a consequence of the standard conjectures. It has beeen proved in some cases : it is true for curves, this case is a consequence of all the work of Faltings on Mordell conjecture.