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I'm an applied model theorist, and open image theorems are important in the mathematical structures I study (they limit the number of types of elements being realised, and therefore keep things model theoretically nice e.g. stable).

So I have some idea as to why these open image theorems should hold from a model theoretic viewpoint, and I know that these are regarded as important theorems, but I don't think I've ever come across a diophantine application of an open image theorem in the literature and I'd like to see one.

I'm most familiar with Serre's open image theorem for elliptic curves so an example in this context would be ideal.

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This is kind of a trivial example, but remember that the open image theorem for the Tate module of G_m is the same as the statement that the cyclotomic polynomial is irreducible, and as we know, that has many applications. –  David Corwin Oct 10 '12 at 14:52
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3 Answers

up vote 5 down vote accepted

Here's an application to independence of Heegner points. (But if you search on MathSciNet for papers that reference Serre's two results, I expect you'll find a very large number of applications.)

Let $E/\mathbb{Q}$ be an elliptic curve with no CM, and let $\Phi:X_0(N)\to E$ be a modular parametrization. (Wiles et.al. show that $\Phi$ exists for all such $E$.) The modular curve $X_0(N)$ has special points called Heegner points associated to pairs $(C,\Gamma)$, where $C$ is a CM elliptic curve and $\Gamma\subset C$ is a cyclic subgroup of order $N$. More precisely, we can associate to each imaginary quadratic field $K$ (satisfying some conditions) a Heegner point $x_K\in X_0(\overline{\mathbb{Q}})$ associated to the maximal order in $K$.

Theorem [1] Let $K_1,\ldots,K_r$ be distinct imaginary quadratic fields such that the odd parts of their class numbers are sufficiently large. Then the points $\Phi(x_{K_1}),\ldots,\Phi(x_{K_r})$ are linearly independent in the group $E(\overline{\mathbb{Q}})$.

The proof uses Serre's image of Galois theorem in a crucial way. Not simply that the image of Galois is open in each $\hbox{Aut}(T_\ell(E))$, but also that it is surjective for almost all $\ell$.

[1] M. Rosen, JH Silverman, On the independence of Heegner points associated to distinct quadratic imaginary fields, Journal of Number Theory 127 (2007), 10-36.

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Well, this isn't explicitly diophantine, but here goes:

If $f$ is a level one weight $k$ eigenform with rational coefficients, the image of the attached Galois representation

$\rho_f:G_{\mathbb{Q}}\rightarrow GL_2(\hat{\mathbb{Z}})$

is open in the subgroup $G$ defined by demanding

$det(G)\subset \hat{\mathbb{Z}}^{\times{k-1}}$.

In particular, the image contains an open subgroup of $SL_2(\hat{\mathbb{Z}})$. This has the following arithmetic consequence:

For almost all prime numbers $p$, there exists a non-solvable Galois extension $K/\mathbb{Q}$ ramified only at $p$.

In fact, Serre shows that for the unique normalized weight 12 level 1 cuspform, the list of exceptional primes is 2,3,5,7,23,691. This theorem is now known for all p, although the last known case, p=7, was resolved only very recently by Dieulefait.

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A Google search revealed that the paper by Dieulefait is arxiv.org/abs/1005.4209. –  Chandan Singh Dalawat Oct 11 '12 at 14:25
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Serre's open image theorem (on page IV-20 in his book "Abelian $l$-adic representations...) for non-CM elliptic curves $E/K$ is equivalent to the statement that, for almost all $l$ (how large depending on $E$ and $K$), the $l$-adic representation attached to $T_l(E)$ is surjective. Ditto for the mod-$l$ representation. These are nice examples of number theoretic applications. Explicit bounds are also known (work of Hall, Cojocaru and others...). Note that how large $l$ must be is expected to be independent of $E$, and should depend only on $K$. For example, if $K = \mathbb{Q}$, it is hoped that 37 is large enough for any non-CM elliptic curve.

It is conjectured that, for $A/K$ any abelian variety for which $End_{\overline{K}}(A) = \mathbb{Z}$, there should be a similar open-image theorem. This is known (by work of Serre) when the dimension of $A$ is 2,6, or odd. In particular, for such an $A/K$, and for sufficiently large $l$, the mod-$l$ representation has image $GSp_{2g}(\mathbb{F}_l)$. This is also nice.

(Work of Bogomolov says that the $l$-adic image of $A$ is open (with the $l$-adic topology) in $G_{A,l}(\mathbb{Q}_l)$ ; here $G_{A,l}$ is the $l$-adic algebraic monodromy group. See this blog post of Martin Orr for a discussion of these groups.)

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Thanks, but what you describe is quite close to a rephrasing of the open image theorem. Maybe I should have put diophantine application in the question. Sorry about that - I'll edit the question –  Adam Harris Oct 10 '12 at 15:05
    
@Adam: Oh I see. Maybe the theorem is somehow used in level lowering of Frey curves attached to putative solutions of Diophantine equations? Maybe it's even used in the proof of FLT? –  Barinder Banwait Oct 11 '12 at 10:16
    
@Barinder:I was hoping for something like that (or maybe naively I was thinking that there would be something that could be explained in 3 mins as background at the start of a talk to justify why the open image theorem is important). However I'm pretty happy with Kevin and Joe's examples –  Adam Harris Oct 11 '12 at 10:28
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