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# Example of a numbertheoreticdiophantine application of an open image theorem

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 number theoretic 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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# Example of a number theoretic application of an open image theorem

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 number theoretic 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.