The weil conjectures can be proved by using some algebraic geometry methods and the establish of etale cohomology. However, can I compute ζ functions over finte field for some simple examples of varieties directly，without using advanced concepts such as scheme and etale cohomology？ Just for
- the projective line
- projective spaces
- elliptic curves
Moreover，what's thier corresponding complex varieties which satisfies the betti number argument.