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!


1 Answer 1


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.

  • $\begingroup$ 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 $E(K)$? $\endgroup$
    – Shreya
    Jun 30, 2019 at 8:37
  • 1
    $\begingroup$ 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. $\endgroup$ Jun 30, 2019 at 11:24
  • $\begingroup$ Ah, thanks a lot! I just realised that I need to read the proof in the generalized case for number fields to get answers of all the whys and how's I had/have. Thanks! $\endgroup$
    – Shreya
    Jun 30, 2019 at 11:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.