Hello,


I remember reading that if $X/\mathbf F\_q$ is a projective smooth global complete intersection, then the characteristic polynomial of the $\mathbf F\_q$-linear Frobenius of $X$
on $H^i\_{et}(X\otimes\overline{\mathbf F\_q},\mathbf Q\_l)$, $l\nmid q$, has integer coefficients and is independent of $l$.


How does one prove this fact?


Thanks



global complete intersection and independence of $l$
If you allow a proof using the Weil conjectures, there's an otherwise simple proof of this for any smooth projective variety in Deligne's first paper proving the Weil conjectures. See pages 276-277 of [Weil I](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1974__43_/PMIHES_1974__43__273_0/PMIHES_1974__43__273_0.pdf).


Edit: Sorry Keerthi! Didn't see your comment when I posted this.


