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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.

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up vote 12 down vote accepted

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.

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