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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.

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Douglas Ulmer has a series of papers on elliptic curves over global function field, e.g. Elliptic curves with large rank over function fields. Ann. of Math. (2) 155 (2002), no. 1, 295–315. You can start by looking there and in the reference therein. –  Valerio Talamanca Apr 8 '11 at 13:18
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3 Answers

up vote 13 down vote accepted

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.

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Definately, this is just what I was after. Thanks! –  Giuseppe Apr 9 '11 at 15:58
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First, let me agree with known google's (ha ha) suggestion.

There is no textbook treatment of the function field case by itself, as the basic theory can be done for both number fields and function fields pretty much the same way. Lang's book "Diophantine Geometry" does that, for example.

The statement about Sha and the Brauer group works more generally, you don't need to assume that the surface is K3. This is not such an elementary result. Ulmer's notes has references.

Ways in which the function field is easier:

A lot is known about BSD. See Ulmer's notes.

An easy example is the function field analogue of Mazur's theorem, which follows almost immediately from the existence of modular curves and the fact that their genus grows.

(Shameless plug) My paper Explicit p-descent for elliptic curves in characteristic p. Compositio Math. 74(1990), 247-258 proves Mordell-Weil and Siegel's theorem for function fields by taking advantage of the existence of a derivation.

A search for "elliptic curves function fields" in mathscinet should provide a lot of other examples.

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It seems truly remarkable to me that proving the list of possible torsion in the number field case is way harder than the function field case; I don't understand Mazur's proof, but I've glanced through it, and it seems on a different planet to the proof in Ulmer's notes for the function field case. –  Giuseppe Apr 9 '11 at 0:50
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(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.

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@Stefano: it looks like a very nice paper. @everyone: please keep the self-advertising coming! –  Pete L. Clark Apr 18 '11 at 1:28
    
@Pete: Thank you very much for your nice words! –  Stefano V. Apr 18 '11 at 8:06
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