Grothendieck-Lefschetz fixed point theorem for nonconstant sheaves? Let $X$ be a variety over ${\bf F}_q$. The Grothendieck-Lefschetz fixed point formula gives
$$|X({\bf F}_q)|=\sum_i(-1)^i\text{Tr}(\text{Frob},H^i_c(X,\bf{Q}_\ell))$$
for the constant $\ell$-adic sheaf on $X$. 
More generally but are there local systems/constructible sheaves on $X$ such that the point counting on the LHS still remain valid?

Unfortunatley I have not enough reputation to add a comment, but you could look at the case $X(\mathbb{F}_q) = \emptyset$ which shows you that such sheaves might exist. Perhaps you should allow $q$ to vary?
