TITLE: Quasi-compact maps in Number Theory
QUESTION [9 upvotes]: Can someone give me an example of a non-quasi-compact morphism of schemes which arises naturally in the field of Algebraic Number Theory?

REPLY [7 votes]: Rex, about your scheme-theoretic image problem, I don't know how "contrived" the following example is (I am afraid it is not particularly related to number theory).
Notations: $R$  is a discrete valuation ring, $t$ a uniformizer, $R_n:=R/(t^{n+1})$ ($n\in\mathbb{N}$), $X_n=\mathrm{Spec}\,R_n$, $A=\prod_n R_n$.
Take $X:=\coprod_n X_n$ and $Y:=\mathrm{Spec}\,A$. There is a natural open immersion $f:X\to Y$ since each $X_n$ embeds in $Y$ as an open and closed subscheme.
The scheme-theoretic image of $f$ is $Y$: since $Y$ is affine, it just means that each $x\in A$ vanishing on each $X_n$ is zero, which is obvious. (In fact, $A=\Gamma(X,\mathcal{O}_X)$).
But $X$ is not topologically dense in $Y$: indeed, consider $x=(t,t,\dots)\in A$. Then $x$ is locally nilpotent on $X$ but not nilpotent on $Y$, hence the open set $D(x)\subset Y$ is nonempty and disjoint from $X$.

REPLY [4 votes]: You might see non-quasi-compact maps in the context of universal covers of maximally degenerate pointed curves, and depending on who you ask, this might be called algebraic number theory.  Specifically, a maximally degenerate positive genus (proper) curve $X$ has the form of a connected graph made out of finitely many projective lines intersecting transversely, where each line has exactly 3 special points (namely intersections and markings).  If $x$ is a marked point, then $\pi_1^{geom}(X,x)$ is a finitely generated free group.  One then has a universal cover $(\tilde{X},\tilde{x}) \to (X,x)$, where $\tilde{X}$ is a tree of projective lines.  The covering map is étale but not quasi-compact.
Gerritzen and van der Put wrote a book (Schottky Groups and Mumford Curves, Springer LNM 817) describing some number-theoretic data like theta functions on these objects.  Some brief web searching suggests that there seem to be some more modern treatments using rigid analytic techniques (that I don't really understand).
I'm afraid this doesn't answer the more focused question about scheme-theoretic image that you posed in the comments.