I am wondering if there is, or is any hope for, a local-to-global proof of the Mordell–Weil theorem. Below I'll specialize to elliptic curves over $\mathbb Q$ for simplicity, but I'm interested in the general case too. Also, to be more precise I am wondering about the proof of the full Mordell–Weil theorem assuming the weak version, but any ideas about variations would welcome too.
It seems that the obstruction in proving the full M.–W. theorem from the weak version is the possibility of divisible elements in $E(\mathbb Q)$. To get some control on this, I want to use the embeddings $E(\mathbb Q)\hookrightarrow E(\mathbb Q_p)$ and the nice "virtual" description of the latter group, namely that it has a subgroup of finite index isomorphic to $(\mathbb Z_p,+)$ (VII.6.3 in Silverman's AEC). The group $(\mathbb Z_p,+)$ has very little divisibility by $p$ in the sense that one has $\bigcap_{n\geq0} p^n\mathbb Z_p=0$, hence the same holds for $E(\mathbb Q)$ (edit: this is not true—see below).
This leads to the following:
Question. Suppose $A$ is an Abelian group satisfying the following properties:
- For all primes $p$, one has $\bigcap_{n\geq0}p^nA=0$.
- For all $m\geq1$, the quotient $A/mA$ is finite.
Does it follow that $A$ is finitely generated? If not, can one deduce a stronger algebraic property of $E(\mathbb Q)$ using the embeddings above that would lead to finite generation?
Edit: We cannot guarantee that $E(\mathbb Q)$ satisfies condition (1) above, since it may have nontrivial torsion. But we can freely pass to the quotient by the torsion, since by VII.3.1 of AEC (injectivity of torsion under reduction) the torsion of $E(\mathbb Q)$ is already known to be finite (or use the Nagell–Lutz theorem?). But it is still not clear to me that we can guarantee $\bigcap_{n\geq0}p^nA=0$ for all primes $p$, only that for each fixed prime there is a finite-index subgroup of $A$ for which this holds. Maybe there is an easier counterexample to this weaker version of the question above.
Also, The resemblance between Mordell's theorem and Dirichlet's unit theorem makes me realize that a proof of the M.–W. theorem along these lines should also prove Dirichlet's unit theorem. This makes me more skeptical that such a proof exists.
