What is etale descent? I have a vague notion that, for example, given a variety $V$ over a number field $K$, etale descent will produce (sometimes) a variety $V'$ over $\mathbb{Q}$ of the same complex dimension which is isomorphic to $V$ over $K$ and such that $V(K)=V'(\mathbb{Q})$. Is this at all right? How does one do such a thing?



What is etale descent?
Let $L/K$ be a Galois field extension and consider a variety $Y$ over $L$. The theory of (Galois) descent addresses the question whether $Y$ can be defined over $K$.
More precisely, the question is: "does there exist a variety $X$ over $K$ such that $Y = X \times\_{Spec(K)} Spec(L)$".


Now assume such $X$ does exist. In this case $Y$ is endowed with $Gal(L/K)$ action coming from an action on the second factor. 


Conversely, if $Y$ has a Galois action compatible with the action on $Spec(L)$, then $Y$ descends to some $X$ defined over $K$. $X$ is actually a quotient of $Y$ by $Gal(L/K)$ (so that the conjugate points glue together to form one point on $X$).
Note that the set of $K$-points of $X$ is the set of Galois fixed points.


Example. $K = \mathbf R$ and $L = \mathbf C$.
For any real variety, the set of complex points admits the action of $
\mathbf Z/2$ by complex conjugation.
Conversely, if a complex variety is endowed with conjugation, it descends to a real variety. This is in fact an exercise in Hartshorne.


Remarks. Theory of descent also classifies all possible $X$'s arising from $Y$. Such $X$'s are called forms of $Y$. They are in 1-1 correspondence with a certain Galois cohomology group.


