On the characteristic polynomial of the Frobenius on \'etale cohomology
  Let $X$ be a smooth proper variety of even dimension $d$ over a finite field. We establish a restriction on the value at $(-1)$ of the characteristic polynomial of the Frobenius on the middle-dimensional \'etale cohomology of $X$ with coefficients in ${\mathbb Q}_l(d/2)$. 
Andreas-Stephan Elsenhans and J\"org Jahnel
math.AG math.NT