I have been wondering recently about the geometric information encoded in the zeta function of a (smooth, projective) variety over a finite field - or in its étale cohomology (i.e. l-adic cohomology) as a Galois module, I hope it's not a blunder to say they are equivalent data, if Tate's conjecture holds. For curves, one can read the genus off of it, but what about other invariants, such as the existence of certain maps to projective space?
A precise question could be if we can tell whether a curve is hyperelliptic by only looking at its zeta function.
I have a hunch this is not the case. If I'm not mistaken, the interesting part of the étale cohomology of a curve is essentially the Tate module of its Jacobian, and the hyperelliptic involution acts on the Jacobian as the inverse, it does not provide an interesting automorphism. Furthermore, if étale cohomology supposedly mimics the usual singular cohomology then I shouldn't expect it to single out hyperelliptic curves - this is a very weak argument, I admit, maybe hyperelliptic curves have peculiar Galois actions. I would expect a counterexample to exist, but a heuristic argument for why it should or shouldn't be true would be great as well.