TITLE: Idea of using etale site
QUESTION [13 upvotes]: I have just read an article which mentions that, when Grothendieck considered using etale morphism, he did borrow the idea from Riemann that multivalued function on an open subset of complex plane should live in Riemann surface covering it instead of the open subset itself.
The question is: how these things are related? Any detailed explanation is very welcome and appreciated. More generally, what are the good properties etale morphism has which make it essential in solving Weil conjecture.I understand that Grothendieck tried to search for a cohomology theory in algebraic geometry which is similar to classical cohomology theory on manifold theory but  I dont know what obstructions are there preventing a cohomology theory in AG behaves similar to one in manifold for which etale site was introduced to overcome them.

REPLY [6 votes]: With regards to the Weil conjectures:
I believe it was known to Weil that, to prove most of the Weil conjectures, a cohomology theory for varieties with coefficients in a field of characteristic zero would just have to satisfy a few properties that are familiar from singular cohomology:

The Lefschetz fixed point formula, applied to the endomorphism Frob_p, and finiteness of Betti numbers immediately give rationality of the zeta function.
Poincare duality, suitably twisted to account for the fact that Frob_p should act on $H^{2n} (X)$ for $X$ an $n$-dimensional smooth projective variety by multiplication by $p^n$, gives the functional equation.
The Riemann hypothesis is the only one I'm not sure about - Deligne's proof of this involves many clever steps that had no antecedent in singular cohomology.
The fact that Betti numbers are constant in smooth proper fibrations suffices for the Betti number conjecture, and a comparison theorem to singular cohomology over the complex numbers, (although Weil may not have had the full picture of an arithmetic family of varieties as being the same as a geometric family as clearly as Grothendieck did) 

So why does etale cohomology have these features? I think experience shows that any cohomology theory we define has these features as long as it satisfies some very basic tests - off the top of my head, I can't think of any cohomology theory that has the right Betti numbers for a curve of genus $g$ that doesn't have all these properties.
$\ell$-adic etale cohomology avoids the trivial failure of all the cohomology theories you can construct with just the Zariski topology - either cohomology of the constant sheaf $\mathbb Z$ in the Zariski topology, or cohomology of some coherent sheaf, probably powers of the cotangent bundle to mimic Dolbeaut cohomology in characteristic zero. The first one has the wrong Betti numbers, and the second one has coefficient field the base field, so has coefficients of characteristic $p$ in characteristic $p$, and so cannot prove the Weil conjectures.