From Academic Kids

In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. This proved successful as a strategy, about a dozen years after the idea was mooted in the early 1960s. This theory is an example of a Weil cohomology theory in algebraic geometry, and as such it continues to play an important role in the more general theory of motives.
The formal definition of étale cohomology is as the derived functor of the functor of sections,
 F → Γ(F),
for a type of sheaf. The sections of a sheaf can be thought of as Hom(Z,F) where Z is the sheaf returning always the integers as abelian group; the sheaf F is understood in the sense of a Grothendieck topology. The idea of derived functor here is that the sheaf of sections doesn't respect exact sequences; according to general principles of homological algebra there will be a sequence of functors H^{i} for i = 0,1, ... that represent the 'compensations' that must be made in order to restore some measure of exactness (long exact sequences arising from short ones). The H^{0} functor coincides with the section functor Γ.
In these very abstract terms, the existence of such a theory comes down to some properties of étale morphisms in scheme theory, allowing us to use étale coverings as a Grothendieck topology, and some further proofs in homological terms, showing for example that injective resolutions are to be found in the sheaf category. To a very great extent, this attitude masks what is going on.
Some basic intuitions of the theory are these:
 The étale requirement is the condition that would allow one to apply the implicit function theorem if it were true in algebraic geometry (but it isn't  implicit algebraic functions are called algebroid in older literature).
 There are certain basic cases, of dimension 0 and 1, and for an abelian variety, where the answers with constant sheaves of coefficients can be predicted (via Galois cohomology and Tate modules).
As it turned out, these base cases in effect determined the theory (perhaps unexpectedly), but the case of a general sheaf on a curve is already complex. Further contact with classical theory was found in the shape of the Grothendieck version of the Brauer group; this was applied in short order to diophantine geometry, by Yuri Manin. The burden and success of the general theory was certainly both to integrate all this information, and to prove general results such as Poincaré duality and the Lefschetz fixed point theorem in this context.
With hindsight, much of the general machinery of topos theory proved unnecessary for a minimal treatment of the étale theory (though applicable to the more subtle crystalline and flat cohomology) — this is Deligne's view as expressed for example in SGA4½. On the other hand, étale cohomology quickly found other applications, for example in representation theory, going beyond the initially planned application.
ladic cohomology groups
In applications to algebraic geometry over a finite field F, the main objective was to find a replacement for the singular cohomology groups, which are not available in the same way as for geometry of an algebraic variety over the complex number field. The hope, which was generally upheld, was that a replacement would be found in the shape of <math>\ell<math>adic cohomology. Here <math>\ell<math> stands for any prime number with
 <math>\ell<math> ≠ p
where p is the characteristic of F. One considers, for schemes V, the cohomology groups
 H^{i}(V, Z /<math>\ell<math>^{k}Z)
and defines
 H^{i}(V, Z_{<math>\ell<math>})
as their inverse limit. Here Z_{<math>\ell<math>} denotes the ladic integers, but the definition is by means of the system of 'constant' sheaves with the finite coefficients Z/<math>\ell<math>^{k}Z.
The reason that one might guess that this leads to the correct definition, is that in the case that V is a nonsingular algebraic curve and i = 1, it can be shown that H^{1} is a free Z_{<math>\ell<math>}module of rank 2g, dual to the Tate module of the Jacobian variety of V, where g is the genus of V. Since the first Betti number of a Riemann surface of genus g is 2g, that value is reassuring. This becomes a kind of 'base case' for inductive study of the general case (that is, i > 1 or V of dimension > 1). It also shows why the condition <math>\ell<math> ≠ p is required: when <math>\ell<math> = p the rank of the Tate module is at most g.
To remove any torsion subgroup from the <math>\ell<math>adic groups (which can occur, and was applied by Mike Artin and David Mumford to geometric questions) the definition
 H^{i}(V, Q_{<math>\ell<math>})
with the <math>\ell<math>adic numbers Q_{<math>\ell<math>} is typically used.
An application to curves
This is how the theory could be applied to the local zetafunction of an algebraic curve.
Theorem. Let X be a curve of genus g defined over the finite field with p elements. Then for every n greater or equal 1 one has
 <math>\#X(\mathbb F_{p^n}) = 1 \sum_{i=1}^{2g} \alpha_i^n+p^n<math>,
where <math>\alpha_i<math> are certain algebraic numbers satisfying <math>\alpha_i=\sqrt p<math>.
Notes
 This agrees with the projective line being a curve of genus 0 and having p^{n}+1 points.
 We see that number of points on any curve is 'rather close' to that of the projective line.
Idea of proof
According to the Lefschetz fixed point theorem, the number of fixed points of any morphism <math>f:X\to X<math> is equals to the sum
 <math>\sum_{i=0}^{\dim X}(1)^i\mathrm{Tr} f_{H^i(X)}<math>.
This formula is valid for ordinary topological varieties and ordinary topology, but it is wrong for most algebraic topologies. However, this formula does hold for étale cohomology (though this is not so simple to prove).
The points of X that are defined over <math>\mathbb F_{p^n}<math> are those fixed by F^{n} where F is the Frobenius automorphism in characteristic p.
The étale cohomology Betti numbers of X in dimensions 0, 1, 2 are resp. 1, 2g, and 1.
According to all of these,
 <math>\#X(\mathbb F_{p^n}) = \mathrm{Tr} F^n_{H^0(X)} \mathrm{Tr} F^n_{H^1(X)} +\mathrm{Tr} F^n_{H^2(X)}<math>.
This gives the general form of the theorem.
The assertion on the absolute values of the αs requires some deeper argument.
The whole idea fits into the framework of motives: formally [X] = [point]+[line]+[1part], and [1part] has something like <math>\sqrt p<math> points.