I will show a few different approaches to the Weak Mordell-Weil theorem and how they are all really the same proof. The various proofs use the Hermite-Minkowski theorem, Class Field Theory and the Dirichlet’s Unit theorem/finiteness of Class group.
Let be an elliptic curve over a number field. The Mordell-Weil theorem says that the group of rational points is finitely generated. This is usually proven in two steps:
- Weak Mordell-Weil Theorem: We prove for some that is finite.
- Theory of heights: We define the notion of a height of a point on (roughly, how many bits of information one would need to store the point). Using this and (1), the completion of the proof is quite formal.
I will focus here on the weak Mordell-Weil theorem and in particular, an approach to it using the Hermite-Minkowski theorem. This approach will apply without very little change to the case of Abelian varieties and the general technique seems to be applicable in great generality.
The idea of the proof is as follows:
- Reduce the case of general to supposing that contains the n-torsion using the Kummer sequence.
- Use the Kummer Pairing to reduce to showing that the inverse image of under the multiplication by map generates a finite extension of .
- Show that there is a smooth, proper model of over an open subset of and hence the inverse image of generates an extension etale over , not just .
- Apply Hermite-Minkowski.