TITLE: Elementary examples of the Weil conjectures
QUESTION [37 upvotes]: 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).

REPLY [6 votes]: Any variety which admits a cell decomposition gives an example. Cell decomposition of $X$ is a stratification $X=\bigcup\limits_n X^n$ such that $X^n\setminus X^{n-1}=\coprod\limits_{i=1}^{k_n}\mathbb{A}^n$ is a disjoint union of affine spaces.
Then $\#X(\mathbb{F}_q)=k_0+k_1q+\dots+k_dq^d$ so zeta-function of $X$ is a product of zeta-functions of affine spaces hence rational.
Examples of such varieties are given by flag varieties of reductive groups(Bruhat decomposition).
More generally, rationality of zeta-functions of moduli spaces  of some objects can be proven elementary by counting these objects over finite field. For example, for quiver varieties it is done by V. Kac in "Root systems, representation of quivers and invariant theory".
Edit: Another, less elementary example is given by rational surfaces. For them Chow groups satisfiy the axioms of Weil cohomology theory, as explained in Manin's book "Cubic forms", so the rationality follows.