Weak Mordell-Weil over number fields

Question

I have a question regarding the Mordell Weil theorem a number field $K$. I read the proof of the Mordell Weil theorem in "rational points on elliptic curves" by Tate and Silverman. They presented a proof for the case where $E[2] \in E (\mathbb{Q}) $ where $E : Y^2 = X(X^2 + AX + B)$ and mentioned before hand that it is possible to prove the case over number fields in the same fashion (thus without the use of group cohomology) with a little help of algebraic number theory.

After a bit of investigating, I concluded that every reasoning also holds for number fields except for the proof that the image of the map $\alpha : E(K) \rightarrow K^* / K^{*2}$ is finite (proposition 3.8(c)). In the proof they claimed that every squarefree integer representing the corresponging quadratic residue class divides $B$.

My reasoning for number fields was as follows:

Let $P = (x,y) \in E$, then $\alpha(P) = x \pmod {K^{*2}}$. Since $K$ is the field of fractions of $O_K$ (ring of integers) we have that $x = \frac{a}{b}$ for $a,b \in O_K$. Now consider $S_P := \{ \rho \in \text{Max}(O_K) \: : \: v_{\rho}(\alpha(P)) \neq 0 \pmod 2 \}$ where $v$ is just the valuation of the prime ideal factorization. My claim is that every prime ideal in $S_P$ must be a divisor of the ideal generated by $B$.

My question: is this correct? And if so, how do I prove this?

Thanks in advance.

Answer 1

That's more or less the right way to do it. The proof of your claim (or something similar) will require (1) finiteness of the ideal class group (or at least, the 2-part of the class group) and (2) finite generation of the group of units in $O_K$, i.e., Dirichlet's unit theorem, although again what's really needed is that $O_K^*/(O_K^*)^2$ is finite. Anyway, you'll find all this in Chapter VIII Section 1 of The Arithmetic of Elliptic Curves. The claim is more-or-less the content of Propositions VIII.1.5 and VIII.1.6, although the specific statement is a bit buried in the proofs of those propositions. There's no group cohomology involved, but in some sense one is looking at 1-cocycles for the step where one goes to an extension field where the 2-torsion is rational; cf. the reduction Lemma VIII.1.1.1.