If an algebraic variety $X$ over a field characteristic p is given by equations $f\_i(x\_1,...,x\_k) = 0$, we can consider the variety $X^{(p)}$ obtained by applying p-th powers to all the coefficients of all $f\_i$'s.
Frobenius morphism, as I understand it, is a morphism $X \to X^{(p)}$,
given on points as raising all coordinates to p-th power. 


Can anyone please explain me, what is the geometric Frobenius, as opposed to the arithmetic one?


EDIT: Thanks to Florian and George for the answers. I understand the difference now.
I accepted Florian's answer because he was first and also because I found the last link 
<http://www.math.mcgill.ca/goren/SeminarOnCohomology/Frobenius.pdf> he provided especially helpful.



Geometric vs Arithmetic Frobenius
Geometric and arithmetic Frobenius live in a Galois group, they are different from the Frobenius morphism. The Galois group of a finite field of cardinality $q$ has a canonical generator $x \mapsto x^q$; this is the arithmetic Frobenius element. Its inverse, i.e., $x \mapsto x^{1/q}$, is the geometric Frobenius element. The Galois group of a non-archimedean local field (i.e., a finite extension of $\mathbb Q\_p$ or $k((x))$ for a finite field $k$) maps surjectively to the Galois group of its residue field (a finite field); an element in the inverse image of an arithmetic/geometric Frobenius is still called arithmetic/geometric Frobenius (but there is no longer a canonical choice).


Finally, I think the reason for the term "geometric" is that for a variety $X/k$ ($k$ a finite field of cardinality $q$), we have a canonical isomorphism $X^{(q)} \cong X$, so the $q$-power Frobenius morphism gives rise to a map $F : X(\bar k) \to X(\bar k)$. The Galois group acts on $X(\bar k)$ as well, and the action of the geometric Frobenius element agrees with $F$.


EDIT: Oops, on $X(\bar k)$ the action of the Frobenius morphism agrees with arithmetic Frobenius, but on the étale cohomology of $X\_{\bar k}$ it agrees with geometric Frobenius. Let me try to find a reference...


Here is one (see p.89). The file name seems to indicate that these are Brian Conrad's, but they are not on his web page as far as I can tell, so I hope he doesn't object to the link! 


<http://math.unice.fr/~dehon/CohEtale-09/Elencj_Etale/CONRAD%20Etale%20Cohomology.pdf>


I think I heard that it was Deligne who coined the term "geometric Frobenius element". Deligne's Bourbaki talk in 68/69 doesn't seem to give it a name. (See Jay Pottharst's translation at <http://math.bu.edu/people/potthars/writings/deligne-l-adic.pdf>, in particular Prop. 4.8.) Deligne mentions SGA 5.XV. I don't have time to check further, I guess it has more on the fact I mentioned but not on the terminology.


<http://www.msri.org/publications/books/sga/sga/5/SGA5-page-454.html>


Finally see Katz's "Review of l-adic cohomology" in the Motives volumes.


<http://books.google.at/books?id=v2CuklFFV5IC&pg=PA26&lpg=PA26&dq=%22geometric+frobenius+element%22&source=bl&ots=QUaysRdc3L&sig=4U_nC8QPWQjdg9RUi1-hHXt1Iec&hl=en&ei=WvstTLzbH8P38Aaj1q2fAw&sa=X&oi=book_result&ct=result&resnum=4&ved=0CBwQ6AEwAw#v=onepage&q=%22geometric%20frobenius%20element%22&f=false>


(scroll back one page)


**Update:** I found some expository notes I couldn't find yesterday. Like Brian Conrad's notes they explain why geometric Frobenius has the same action as the Frobenius morphism on étale cohomology. (They use the terminology of arithmetic/geometric Frobenius morphism though.) 


<http://www.math.mcgill.ca/goren/SeminarOnCohomology/Frobenius.pdf>


