How explicit is "explicit"?  Even norms of field extensions are somewhat inexplicit.  

The following gives just *one* example, but the method extends to other examples.  If you have a pencil of degree $d$ hypersurfaces in $\mathbb{P}^n$ over $\mathbb{F}_q$, if you can count the number of points in the base locus, and if you can count the number of (rational) singular members of the pencil and the number of points of each singular fiber, then you can sometimes prove that the number of points in *some* (rational) smooth member of the pencil cannot be congruent to $1$ modulo $q$.  For instance, if $q=2^r$ for $r\geq 2$, this applies to the pencil from the following: http://mathoverflow.net/questions/115361/when-is-the-kernel-of-the-etale-fundamental-group-in-a-fibration-abelian/115498#115498  

That example is a pencil of degree $4$ plane curves, the base locus has $4$ rational points, there are precisely two singular fibers, both rational, one of which has $4q-2 = (4q-6)+4$ rational points, and the second of which has $q+1 = (q-3)+4$ rational points.  The union of the singular fibers has $4+(4q-6)+(q-3) = 5q-5$ rational points.  Thus, the number of rational points contained in no singular fiber is $$(q^2+q+1)-(5q-5) = q^2-4q+6.$$ There are $q-1$ smooth fibers.  If the number of points in a smooth fiber is congruent to $1$ modulo $q$, then the number of non-base points is congruent to $1-4=-3$ modulo $q$.  Modulo $q$, the total number of non-base points in smooth fibers is congruent to $$-3(q-1) = -3q+3.$$  Since $q=2^r$, $6$ is not congruent to $3$ modulo $q$.  Thus, at least one smooth fiber violates the congruence.