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.



Example of a diophantine application of an open image theorem
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.


