Geometric vs Arithmetic Frobenius 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.
 A: Brion & Kumar ["Frobenius splitting methods in geom. and rep. thy" Birkhauser 2005] call the absolute
Frobenius endomorphism the mapping $F_{abs}:X \to X$ which is the identity on $X$ and with
comorphism given when 
$X = \operatorname{Spec}(A)$ is affine by the rule $(f \mapsto F_{abs}^*(f) = f^p):A \to A$.
This is not a morphism "over $k$" since $F^*:A \to A$ is "semilinear" for the Frobenius
endomorphism of $k$ (= Frobenius automorphism in Galois group of $k$ if $k$ is perfect).
In Jantzen ["Representations of algebraic groups", 2nd edition] 9.1 and 9.2, he describes the absolute Frobenius map - it is "the same" as the one describe by B&K, except that
the codomain is "twisted" to make $F$ a morphism over $k$. For $X = \operatorname{Spec}(A)$
this twisting amounts to: $X^{(p)} = \operatorname{Spec}(A^{(p)})$ where
the $k$-algebra $A^{(p)}$ is $A$ as a ring but an element $a \in k$ acts on $A^{(p)}$ as
$a^{p^{-1}}= a^{1/p}$ does on $A$.
Geometric and arithmetic Frobenii have meaning only (I believe) when $X$ is "defined
over" a finite field; here I'll assume $X$ is defined over $\mathbf{F}_p$. And I'll even suppose $X$ arises by base change to $k$ from the affine $k_0 = \mathbf{F}_p$-scheme $X_0 = \operatorname{Spec}(A_0)$ (otherwise, patch!). 
Then $X = \operatorname{Spec}(A)$ where $A = A_0 \otimes_{k_0} k$.
The arithmetic Frobenius map on $X$ is the $k_0$-morphism $F_{arith}:X \to X$ whose
comorphism is given by $$(f \otimes a \mapsto f \otimes a^p):A = A_0 \otimes_{k_0} k \to A = A_0 \otimes_{k_0} k$$ for $f \in A_0$ and $a \in k$.
Thus the set of $k_0$-points $X_0(k_0)$ is the set of points in $X(k)$ fixed by the
arithemtic Frobenius $F_{arith}$; i.e. the action of $F_{arith}$ on points just gives the
"usual" action of the Frobenius element of the Galois group on rational points (here I must be supposing $k$ to be perfect...)
The geometric Frobenius of $X$ is the $k$-morphism $F_{geom}:X \to X$ whose comorphism
is given by $$(f \otimes a \mapsto f^p \otimes a):A = A_0 \otimes_{k_0} k \to A = A_0 \otimes_{k_0} k.$$ If you pick an embedding $X \subset \mathbf{A}^N$ defined
over $k_0$, then $F_{geom}$ is given on $k$-points in these coordinates by
$$F_{geom}(x_1,\dots,x_N) = (x_1^p,\dots,x_N^p)$$.
The arithmetic and geometric Frobenius are defined (briefly) in Jantzen (loc. cit.). 
Note that $F_{arith} \circ F_{geom} = F_{geom} \circ F_{arith}$ is the "absolute Frobenius" of B&K mentioned above.
Also see Milne's "Lectures on Etale Cohomology" 29.11 for some discussion reconciling
the number theorists with their action of the Frobenius automorphism $\phi=(x \mapsto x^p)$ on the Tate group $T_\ell E$ of an elliptic curve defined over $k_0$ and the algebraic geometers with their action of $F_{geom}$ on $H^1(E,\mathbf{Z}_\ell)$.
A: 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
