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.
Best Answer
Douglas Ulmer wrote up expository notes for his short course at PArk City on precisely this topic:
http://arxiv.org/abs/1101.1939
This might be a good place to start.