Simply connected quasi-projective varieties in positive characteristic I am looking for examples of non-projective (quasi-projective) varieties $X$ defined over a field of positive characteristic, which have trivial étale fundamental group.
It is well known that the étale fundamental group in positive characteristics is a very difficult object, especially so in the non-projective case due to possibly wild ramification at infinity. I'm not even sure if there are examples of the kind above. Is this known?
 A: Here's a remark. It's a generalisation of a theorem of Katz-Lang given by Szamuely and Spieß that for a quasi-projective variety over a perfect field $k$, the abelianised tame fundamental group sits in the following exact sequence
$0\rightarrow T \rightarrow \pi_1^{t,ab}(X) \rightarrow T(Alb_X)\rightarrow 0$
where $T$ is a group related to the torsion subgroup in the Neron-Severi group $NS(X)$ and $T(Alb_X)$ is the full Tate module of the generalised Albanese variety of $X$. The definition of tame covers tries to control exactly this wild ramification at infinity (i.e control the function field for the points in $\mathfrak{X}\setminus X$ where $\mathfrak{X}$ is the corresponding projective). You can find more in Szamuely's new book for example.
A: It is a direct consequence of Abhyankar's Conjecture (which was proved by Raynaud and Harbater) that if $k$ is an algebraically closed field of characteristic $p > 0$, then no affine curve $X_{/k}$ has trivial etale fundamental group.  (Note that for curves, affine = quasi-projective, non-projective, by Riemann-Roch.)
I have some lecture notes on this subject from years back:
http://alpha.math.uga.edu/~pete/fundincharp.pdf
(In contrast to what it says on the first page, they are from 2002.)
Addendum: The comments above give plenty of examples of non-projective quasi-projective varieties with trivial etale fundamental group in characteristic $p$ (or really in characteristic quelconque).  An interesting question left open by these examples is whether there are any (nontrivial) simply connected affine varieties in characteristic $p$.  As I have said, the answer is "no" in dimension one.
A: This is an answer to Pete's question on simply connected affine varieties (I can not put it in a comment because of space limitation). 
I think that in positive characteristic $p$, no affine irreducible variety $X$ of positive dimension is simply connected. We can assume $X=\operatorname{Spec}(A)$ integral because $\pi_1$ is insensible to nipotent elements (SGA IX.4.10). Let $k[t_1,\ldots, t_d] \subseteq A$ be a finite extension with minimal degree $[k(A):k(t_1,\ldots, t_d)]$. Consider the étale cover $Y\to \mathbb A^d_k= \operatorname{Spec}(k[t_1,\ldots, t_d])$ defined by $s^p-s=t_1$. Then $X\times_{\mathbb A^d_k} Y\to X$ is an étale cover of degree $p$. As $k(Y)$ and $k(X)$ are linearly disjoint over $k(t_1,\ldots, t_d)$ ($k(Y)$ is Galois over $k({\bf t}):=k(t_1,\ldots, t_d)$ and $k(Y)\cap k(X)=k({\bf t})$), the tensor product $k(Y)\otimes_{k({\bf t})} k(X)$ is a field. This implies that $X\times_{\mathbb A^d_k} Y$ is connected. 
A: $\mathbb{P}_k^1$, over an algebraically closed field, is such an example. You can adapt the proof that $\mathbb{Q}$ has no unramified extensions to show that $k(t)$ has no unramified extensions.
