# Weak Mordell-Weil for EC using Chevalley-Weil theorem

I am reading the book Applications of Diophantine Approximation to Integral Points and Transcendence by Zannier and Corvaja and, after their proof of the Chevalley-Weil theorem, in Example 3.8 they suggest that it is possible to prove the weak Mordell-Weil theorem using Chevalley-Weil. Honestly, I can't see that, but I am very curious. Does anyone have an idea about it?

The multiplication-by-$$m$$ map $$[m]:E\to E$$ is unramified, so there exists a finite set of primes $$S$$, depending only on $$E$$ and $$m$$, so that for every $$P\in E(K)$$, the field generated by the coordinates of the points in $$[m]^{-1}(P)$$ is unramified outside of $$S$$. (This is where we use Chevelley-Weil.) The degree of that extension is also bounded as a function of $$m$$. It follows from standard results in algebraic number theory that there are only finitely many such fields. Hence $$[m]^{-1}\bigl(E(K)\bigr)$$ is contained in $$E(L)$$, where the extension $$L/K$$ is a finite extension.
I'll let you read elsewhere the fact that it suffices to prove the weak Mordell-Weil theorem under the assumption that $$E[m]\subset E(K)$$. Under that assumption, and with notation as in the first paragraph with $$L=K\bigl([m]^{-1}\bigl(E(K)\bigr)\bigr)$$, there is a well-defined injective homomorphism $$E(K)/mE(K) \longrightarrow \operatorname{Hom}\bigl(\operatorname{Gal}(L/K),E[m]\bigr)$$ defined by $$P \longmapsto \bigl(\sigma\mapsto \sigma(Q)-Q\bigr) \quad\text{for any choice of Q\in E(L) satisfying [m](Q)=P.}$$ Since $$L/K$$ is a finite extension from Chevalley-Weil, and since $$E[m]$$ is a finite group, this gives the finiteness of $$E(K)/mE(K)$$.