TITLE: Elliptic Curves over Global Function Fields
QUESTION [9 upvotes]: I am currently thinking about a problem, and I feel that by knowing more about elliptic curves over extensions of $\mathbb{F}_q(T)$, for $q$ a power of $p$ say, might lead to insight. I am also inspired by the following sentence of Silverman in his Advanced Topics (introduction to Chapter 3):

"Thus conjectures about elliptic curves defined over $\mathbb{Q}$ are often first tested and proven in the easier setting of elliptic curves over $\mathbb{F}_q(T)$."

Unfortunately I can't seem to find a source that deals with this theory systematically, something that studies properties of these curves particular to these global function fields. I put this into Google, and I learned a few nice things, such as, if the corresponding elliptic surface is $K3$, then SHA can be identified with the Brauer group of the surface, but I'm looking for a more text-book or expository type approach that starts at a lower level, say with torsion subgroups, isogenies, Heights and Mordell-Weil Theorem... (and it would be nice if that cohomological fact were also proved therein). Does anybody have any good suggestions?
May I also ask for other ways such curves are 'easier', or better understood, than the 
characteristic 0 case?
I apologise in advance if this question is unfit for the site; I've not been a member for very long.

REPLY [4 votes]: (Warning: another piece of self advertising) Perhaps you might wish to take a look at this paper of mine (joint with A. Bandini and I. Longhi). With the final aim of giving an alternative proof of a classical result of Igusa describing the image of Galois representations attached to non-isotrivial elliptic curves over a global function field $F$ (of characteristic $p>3$), we explain various auxiliary results in the arithmetic theory of such elliptic curves. In particular, using basic properties of Faltings heights of elliptic curves, we give a detailed proof of the function field analogue of a well-known theorem of Shafarevich about F-isomorphism classes of "admissible" elliptic curves over F.
Since we tried to be as elementary as possible, this article might also be viewed as an introduction to various basic topics in the arithmetic of elliptic curves over global function fields (isogenies, Frobenius morphisms, Faltings heights, Shafarevich's theorem, ...) for which we could not find a convenient reference in the literature.