Let an elliptic curve be given.

As the title says I want to know if we can show that two independent points $P$ and $Q$ are generators of the torsion free part of $E$.

For instance let $E:y^2=x^3-17x$. Then I determined with the method from "Rational Points on Elliptic Curves" from Silverman and Tate, that the $E$ has the independent integer points

$$P=(-1,4)\text{ and } Q=(9,24)$$

and rank$(E(\mathbb{Q}))=2$.

How can I show that $\langle (0,0),P,Q\rangle =E(\mathbb Q)$?

  • 1
    $\begingroup$ The generators of the free part are $P_1 = (-4, 2)$ and $P_2 = (-1, 4)$ and $P_1 - P_2 = Q = (9, 24)$. According to Sage. Apart from that you can use the E.saturation method in Sage, to check if a set of points generate the full group. $\endgroup$ Nov 13 '15 at 15:27
  • $\begingroup$ @JesperPetersen I actually was looking for a method so that I am able to check this "by hand". So I guess this is "not possible" because of the high running time of algorithms? I hoped that its atleast possible for simple curve like the one above. $\endgroup$
    – Benjamin
    Nov 14 '15 at 12:49
  • 1
    $\begingroup$ I feel convinced that the general question of determining if $r$ known points generate the full group cannot be down by hand or even by using a computer in reasonable time. To understand if it is possible for your particular curve, my best advice is to read the answer by Silverman and Elkies below and maybe compare with section 3 (or specifically 3.5) of John Cremona's book "Algorithms for Modular Elliptic Curves" (homepages.warwick.ac.uk/~masgaj/book/fulltext/index.html). I believe that the saturation method of Sage originates from Cremona. $\endgroup$ Nov 14 '15 at 14:41
  • $\begingroup$ @JesperPetersen I went through sage (I never used it until now). And I was able to determine the rank, integer points,... of some elliptic curves. But E.saturation did not work for me. Could you explain me (or give a manual, if there exists one) which explains me how to use this to check if two points are generating the Mordell-Weil group? $\endgroup$
    – Benjamin
    Nov 18 '15 at 0:25
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    $\begingroup$ Please, don't feel stupid. I now realize that I am wrong, and have caused too much confusion. For $m,n \in \mathbb{Z}$ we have that $mP_1+nQ = (m+n)P_1 - nP_2$. So for a fixed $n$ we can choose $m$ accordingly, and therefore generate the same points as if we used $P_1$ and $P_2$ as generators. Thus, $P_1$ and $Q$ are generators as well. The generators are not uniquely determined. Sorry, for the confusion. $\endgroup$ Nov 19 '15 at 17:48

Your question asks "How can I show ...". As Jesper Petersen notes, if by that you mean "How can I show by using a software package", then one can use Sage (or undoubtedly Magma or mwrank, etc.) But if you mean "How do I understand how one shows...", that's different. The answer is that the standard descent argument shows that if $P_1,...,P_r$ generate $E(\mathbb Q)/2 E(\mathbb Q)$, for example, then there are explicitly computable constants $C_1$ and $C_2(E)$ so that the effectively computable set $$ \{P\in E(\mathbb Q) : h(P)\le C_1\max h(P_i) + C_2(E)\}$$ contains a set of generators for $E(\mathbb Q)$. So one computes this set, gets a finite set of points, and whittles it down to a generating set. A nice observation (of Don Zagier, I believe) is that if one uses canonical heights, then $$ \{P\in E(\mathbb Q) : \hat h(P)\le 2\max \hat h(P_i)\} $$ (or something like that) contains a set of generators for $E(\mathbb Q)$. This is an exercise on lattices and positive definite quadratic forms. Then one can use estimates relating $\hat h$ and $h$ to search for a set containing generators. This is the basic method. Various ideas can be used to speed the process, but ultimately there is going to be a search over a finite set of potential generators.

  • $\begingroup$ Shouldn't the canonical-height bound depend on the rank? If the Mordell-Weil lattice $L$ looks like the span of ${\bf Z}^r$ and $(1/3, 1/3, \ldots, 1/3)$, then the unit vectors generate $L/2L$, but any set of $\bf Z$-generators must include at least one point of height $\geq r/9$. $\endgroup$ Nov 13 '15 at 18:06
  • $\begingroup$ @NoamD.Elkies Good point. I did put in the "or something like that" to indicate that I wasn't sure I was recalling the result correctly. Maybe it is $\hat h(P)\le \sum \hat h(P_i)$? Anyway, once one knows to look for a result like this, it shouldn't be hard to figure out both the statement and the proof. $\endgroup$ Nov 13 '15 at 19:15
  • $\begingroup$ @JoeSilverman actually I know that generators of $E(\mathbb Q)/2E(\mathbb Q)$ when I know the generators of $\alpha(\Gamma)$ and $\overline{\alpha}(\overline{\Gamma})$ right? (This notation comes from "Rational points on elliptic curves"). So for the elliptic curve given in the title I should know the generators of $E(\mathbb Q)/2E(\mathbb Q).$ and I must be able to calculate the constants from your answer. Is there a method so that I can do this "by hand"? Thanks in advance and thanks for your answer :) $\endgroup$
    – Benjamin
    Nov 18 '15 at 0:22

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