[Math] Elementary examples of the Weil conjectures

Question

I'm looking for examples of the Weil conjectures—specifically rationality of the zeta function—that can be appreciated with minimal background in algebraic geometry. Are there varieties for which one can easily calculate the numbers of points over finite fields and witness the rationality directly? Of course, there are entirely straightforward examples coming from projective space and Grassmannians (or anything with a paving by affines).

Answer 1

Weil himself verifies the conjectures "by hand" for diagonal hypersurfaces, that is, hypersurfaces defined by an equation of the form

$$a_0x_0^{n_0}+a_1x_1^{n_1}+\cdots+a_kx_k^{n_k}=b.$$

The argument is pretty elementary--it essentially uses only character theory. It seems to me likely that this argument heavily influenced Dwork's original proof of rationality.

The paper is quite readable; I learned of it from Akshay Venkatesh.