what does a "rational structure" mean in algebraic group theory For an algebraic group $G$ and a finite field $\mathbb F_q$, what does an "$\mathbb F_q$-rational structure of $G$" mean? Is it always related to a Frobenius map?
I encountered this while reading some books. I have ntried but found no clear and standard definition for this, so I posted this question.
Thanks to everyone.
 A: Hard to say exactly without the context, but it probably means the structure of a variety defined over $\mathbb{F}_q$. Presumably $G$ is already defined over an extension $\mathbb{F}$ of $\mathbb{F}_q$. Then a $\mathbb{F}_q$-rational structure for $G$ is an algebraic group $G_0$ defined over $\mathbb{F}_q$ such that after extension of scalars to $\mathbb{F}$ it becomes isomorphic to $G$. It's "a structure" and not "the structure", because there could be another algebraic group $G_1$ defined over $\mathbb{F}_q$ which also becomes isomorphic to $G$ (and so $G_0$) over $\mathbb{F}$, but which is not isomorphic to $G_0$ over $\mathbb{F}_q$. In that case $G_1$ is a different $\mathbb{F}_q$-rational structure for $G$.
A rather old-fashioned way to define varieties, especially in the context of algebraic groups, is to define a variety $V$ over an algebraically closed field $\overline{K}$ to begin with, then assert $V$ has a structure over a smaller field $K$. This is convenient because you can identify $V$ with a concrete set of points, and for example specify morphisms explicitly by polynomials without having to worry about scheme structures and functorial points.
How much the Frobenius comes into it depends on the situation, but over finite fields it's usually not very far away. It's especially likely to be involved if one is interested in $\mathbb{F}_q$-rational points. To check whether a point with coordinates in $\overline{\mathbb{F}}_q$ is $\mathbb{F}_q$-rational one just verifies that it's fixed by the Frobenius map $x \mapsto x^q$. If the equations defining $V$ are fixed by this map, then $V$ has a $\mathbb{F}_q$ structure.
Often you have $V$, defined over $\mathbb{F}=\overline{\mathbb{F}}_q$, and you know how the absolute Galois group $\mathbb{F}/\mathbb{F}_p$ acts on the points of $V$. Whether $V$ has a $\mathbb{F}_q$-rational structure or not, is the topic of Galois-descent and is slightly subtle.
Milne's algebraic geometry notes deals with all this in the language of $K$-structures:
http://www.jmilne.org/math/CourseNotes/AG.pdf
(Chapter 16 introduces K-structure and deals with descent theory)
