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I've recently started to study elliptic curves and soon enough I came across the Mordell-Weil theorem.

It says, in particular, that if $E: y^2=f(x)$ is an elliptic curve with coefficients in $\mathbb{Q}$, then the set $E(\mathbb{Q})$ of rational points in $E$ is a finitely generated abelian group.

I agree it's an impressive result, but I still can't see any real use for it.

By "real use" I don't mean real-world applications, I'm actually interested in pure, conceptual math: what kind of concept or object does it help me to understand? What problems does it help me to solve? What kind of claim does it help me to prove? Or is it more helpful to disprove things?

I have some background in algebraic number theory, commutative algebra and basic algebraic geometry, so I might be able to understand examples from those areas.

Any insights or references will be helpful, thanks!

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The Mordell-Weil Theorem has many applications in modern number theory, and also to applied number theory, i.e., to cryptographic systems, see for example this article.

For the theory of elliptic curves over $\mathbb{Q}$, Mordell-Weil is essential to define the rank. This rank is very mysterious and there is a famous conjecture attached to it - the Birch-Swinnerton-Dyer conjecture.

More generally, we may also ask for rational points on algebraic curves in general, not just elliptic curves. Here Mordell made also a conjecture that a curve of genus greater than $1$ over the field $\mathbb{Q}$ of rational numbers has only finitely many rational points. In $1983$ it was proved by Gerd Faltings.

Moreover, Mordell's conjecture/Falting's Theorem is a consequence of the abc-conjecture.

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    $\begingroup$ Finiteness of rational points on higher-genus curves is not Mordell--Weil; that is Faltings' solution to Mordell's conjecture. $\endgroup$ – Nefertiti Aug 23 '17 at 8:30
  • $\begingroup$ @Nefertiti My god, of course the Mordell conjecture, I am sorry. $\endgroup$ – Dietrich Burde Aug 23 '17 at 9:05
  • $\begingroup$ I think God will forgive you. :) $\endgroup$ – Nefertiti Aug 23 '17 at 10:52
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I wouldn't really see Mordell-Weil theorem as a tool for proving or disproving something. In every branch of math there are some fundamental and usually very broad questions that motivate the whole research in the field. In number theory (or arithmetic geometry), one of these fundamental questions is: take an algebraic variety over $\mathbb Q$; is it possible to find all of its rational points? Of course we are very far from answering this question in full generality, but Mordell-Weil gives you a good piece of an answer in the very specific case of elliptic curves. As mentioned in the answer above, elliptic curves are the most interesting case among smooth curves, since those of genus $>1$ have only finitely many rational points while those of genus $0$ are either $\mathbb P^1$ or have no rational points. MW theorem is a structure theorem for the remaining case, and I think this already makes it deeply interesting.

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