Can someone help clear up some confusion regarding the proof of Mordell-Weil theorem?

I'm reading the proof(s) of Mordell-Weil theorem using various texts. This post is to make sure what I'm reading and understanding is correct..

Rational points on Elliptic Curves by Silverman Tate gives a proof of Mordell 's theorem when elliptic curves are over rationals, $$\mathbb{Q}$$and have at least one rational $$2$$-torsion point.

Then I noticed that Lawrence Washington's book on Elliptic curves also discuss this result this, first when all roots of cubic $$f(x)$$ are in $$\mathbb{Q}$$ and then just one of them is in $$\mathbb{Q}$$. And the same goes for Cassels' and Knapp's book as well.

Then Lawrence Washington gives a remark that same can be said (i.e., Mordell-Weil holds) when we consider elliptic curves ($$y^2 = f(x)$$) over any number field $$K$$ and $$f(x)$$ has all roots in $$K$$ as we can prove Weak Mordell Weil in this case with some modification of the usual $$\mathbb{Q}$$ argument. But then it says nothing about the case when we just assume that $$f(x)$$ has at least one zero in $$K$$.

Now my questions,

(1) I'm assuming that we can give a proof of Mordell-Weil when elliptic curves are over K and $$f(x)$$ has (just) at least one point in $$K$$. And I think Silverman Tate's proof in case of $$\mathbb{Q}$$ can be generalized, mainly Weak Mordell-Weil's (Lemma $$3.4$$ in the book)

(2) What happens when elliptic curves ($$y^2=f(x)) E(\mathbb{Q})$$ and $$E(K)$$ have no $$2$$-torsion point or $$f(x)$$ has no root in $$\mathbb{Q}$$(respectively $$K$$) ? Is it the situation that is discussed in the J.S.Milne's Elliptic Curves book or J. Silverman's The Arithmetic of Elliptic Curves?

I would appreciate if someone could help answer some or all of the questions above or maybe just refer to an appropriate source to find these proofs/ discussions.

Thank you!

If $$E$$ is an elliptic curve defined over $$K$$ (a number field) and if $$E(K)$$ has a 2-torsion point $$T$$, let $$E'=E/\langle T\rangle$$ be the isogenous curve, let $$f:E\to E'$$ be the isogeny, and let $$f':E'\to E$$ be the dual isogeny. Then you always get maps $$E(K)/f(E'(K)) \to K^*/{K^*}^2\quad\text{and}\quad E'(K)/f'(E(K)) \to K^*/{K^*}^2$$ whose images are effectively calculable subgroups of $$K^*/{K^*}^2$$. However, computing these subgroups (and even proving that they are finite) requires knowing that the ring of integers of $$K$$ has finite class number and finitely generated unit group. So that's why people often take $$K=\mathbb{Q}$$, since then there's no issue with non-principal ideals or non-trivial units. (To be precisely, for this argument one only needs to know that the 2-part of the class group is finite.)
If $$E$$ does not have a $$K$$-rational 2-torsion point, you can go to an extension field where it does, which will be a cubic extension of $$K$$; but then, of course, the unit group and class group may increase.
The computations are simpler, in some ways, if all of the $$2$$-torsion of $$E$$ is $$K$$-rational, since then one can ignore 2-isogenies and simply use a map $$E(K)/2E(K) \to K^*/{K^*}^2 \times K^*/{K^*}^2$$ whose image lies in a finite, effectively computable, subgroup.
• Thanks for your helpful answer. But how about height descent arguments in the case when we are considering elliptic curves over number field $K$ having one or all $2$- torsion points $K$- rational? I assume that they remain the same, but then how do we define height of a point on $E(K)$? Jun 30, 2019 at 8:37
• Roughly, one wants $h(x)$ to be the complexity of the number $x$, i.e., the number of bits it takes to describe $x$. The usual version is called the Weil height and is defined by $$h(x)=\sum_{v\in M_K} \log\max\{ \|x\|_v,1\},$$ where the sum is over a set of normalized absolute values on $K$. Then for a point $P\in E(K)$, one defines $h(P)=h(x(P))$. One shows that $h(2P)=4h(P)+O(1)$, $h(P+Q)\le2h(P)+2h(Q)+O(1)$, and $\{P\in E(K):h(P)\le c\}$ is finite for all $c$. Then the descent argument works. You'll find this in any standard text, including my AEC. Jun 30, 2019 at 11:24