Define an arithmetic scheme $X$ to be a separated, integral scheme, flat and finite type over $\mathbb{Z}$. I am interested in obtaining examples of finite étale covers of arithmetic schemes. I am mainly interested in the case of arithmetic surfaces, but I would enjoy examples of finite étale covers of arithmetic schemes of any dimension $>1$.


Let us restrict to schemes proper over $\mathbb{Z}$ to rule out the possibility of forming a finite integral extension and then restricting to an open subscheme on which the morphism is étale. Let us also restrict to geometrically connected (generic fiber?) to rule out the possibility of base ring extensions.


In the case where $X\to Spec(\mathbb{Z})$ is also smooth, the Katz-Lang finiteness theorem tells us that there will be at most finitely many abelian étale covers of $X$.


Abelian varieties can be a convenient way to create covers: if $A$ is an abelian variety, and $H$ a finite subgroup, then the dual isogeny to $A\to A/H$ is an étale cover of $A$. However, there are no abelian varieties over $Z$. I'm not sure whether this approach could be modified to produce an example.


It would be preferable to stick to regular or at least normal schemes, but that's not strictly necessary.


So, can anyone provide some nice examples of étale covers satisfying these conditions? If you would like to weaken some of the conditions to provide an interesting example (particularly, allowing nonproper or working over another $\mathcal{O}\_K$), I guess I'll allow it.


To summarize, the optimal conditions on $X$ are regular, geometrically connected, proper over $\mathbb{Z}$.


Thanks.



Examples of étale covers of arithmetic surfaces
You can, of course, use Bertini's theorem to make examples of a finite, flat, Galois extension with arbitrary finite Galois group $\Gamma$. Let $M$ be $\mathbb{Z}[\Gamma]$, the group ring of $\Gamma$ with coefficients in $\mathbb{Z}$. This is a finite, free $\mathbb{Z}$-module. Let $r>0$ be a positive integer. Then $M^{\oplus r}$ is also a finite, free $\mathbb{Z}$-module with a natural action of $\Gamma$. Form $S=\mathbb{Z}[M^{\oplus r}]$, the $\mathbb{Z}\_{\geq 0}$-graded polynomial ring over $\mathbb{Z}$ such that the first graded piece $S\_1$ equals $M^{\oplus r}$. Then $\mathbb{P}:=\text{Proj}(S)$ is a projective space over $\mathbb{Z}$ that has a natural action of $\Gamma$. 


Denote by $U\subset\mathbb{P}$ the maximal open subscheme on which $\Gamma$ acts freely. Denote by $F$ the complement of $U$. The fibers of $F$ (over $\text{Spec}\ \mathbb{Z}$) have codimension at least $r$ inside the fibers of $\mathbb{P}$. Hence, assume $r$ is sufficiently large (say $r\geq 2$) so that the fibers of $F$ have codimension $\geq 2$. Denote by $q:\mathbb{P} \to Q$ the quotient of the action of $\Gamma$ on $\mathbb{P}$. This morphism is finite, and it is flat when restricted to $U$. The quotient $Q$ is a projective scheme, say $Q\subset \mathbb{P}^N\_{\mathbb{Z}}$. By Bertini's Theorem, for sufficiently large integers $d$, there exist degree $d$ hypersurfaces in $\mathbb{P}^N\_{\mathbb{Z}}$ whose intersection with $Q$ is a regular, $2$-dimensional scheme $X$ that is disjoint from the image of $F$. Define $\widetilde{X} = \mathbb{P}\times\_Q X$. Then $\widetilde{X}\to X$ is a finite, flat, étale morphism that is Galois with Galois group $\Gamma$.


