@QiaochuYuan: The statement about point counting is incorrect: consider the multiplication by 'n' map on an elliptic curve E with fibre E[n] over some finite field F_q with n invertible in F_q. By passing to an extension, you may assume E[n] =~ (Z/n)^2, and then the formula you suggested in clearly incorrect (unless n=1).
@QiaochuYuan: The statement about l-adic cohomology (that the Leray spectral sequence degenerates), however, is indeed correct, and follows from the result of Deligne mentioned by Dylan Wilson (at least in the projective case, and the proper case is OK too, by the Weil conjectures, though presumably there is a direct argument)