Elliptic Curves - Identifying a Torsion Point, with some Rank thrown in

Question

I'm a totally amateur mathematician who discovered elliptic curves this past summer and am quite fascinated by them. I am trying to learn more about them, but I am hampered by having studied Abstract Algebra 35 years and not remembering such things as what a field or Galois Theory is. I'm trying to fill in gaps by YouTube videos, but it is slow-going now. I do know what a group is and in particular what an abelian group is, but that's about all I understand in some of the highly technical materials I've come across.

I have read about something called a Torsion Point. However, almost everything I find about them throws in all of these advanced math concepts such as fields. But instead of coming right out here and asking all of you for a plain English definition of a Torsion Point, please let me ask my own question and see if I have the right idea.

The elliptic curve in question is $y^2=x^3-3x+34$. (Correction of 18 to 34 in that equation - I copied the wrong one from my source document.) The points that are both rational and integers are $(-3,4)$, $(-3,-4)$, $(-1,6)$, $(-1,-6)$, $(2,6)$, $(2,-6)$, $(5,12)$, $(5,-12)$, $(15,58)$, $(15,-58)$, $(29,156)$, and $(29,-156)$. I found that if I started with $P:(5,12)$, the next points became $2P:(-1,6)$, $3P:(-3,-4)$, $4P:(2,-6)$, $5P:(29,-156)$, $6P:(15,58)$. At this point, the multiples of $P$ cease being integers and probably continue infinitely.

Here's my question: is the starting point $P:(5,12)$ the Torsion Point of this elliptic curve? I happened upon it as the first one of the sequence of integer points by trial and error, but would there be an easier way to find it? Also, would this elliptic curve be considered to have rank $= 1$, since it appears that all of the rational points turn up with just one starting point? I have seen some examples of elliptic curves with rank given, but again, sometimes the language of Abstract Algebra and other advanced math disciplines gets in the way of understanding how this is computed.

Thanks for any plain English help that anyone can give me on this.

Answer 1

If you understand what $nP$ means, then here is what torsion means: a point $P$ is torsion if some multiple $nP$ of it is equal to the identity (that is, the point at infinity). It is a highly nontrivial fact which was proven by Mazur that for elliptic curves over the rationals, to check that $P$ is not torsion it suffices to check that none of the multiples $2P, 3P, 4P, \dots 12P$ are the point at infinity. So if you compute these multiples for your point $P$ and none of them are the point at infinity then $P$ is not torsion.

An elliptic curve over $\mathbb{Q}$ has rank $1$ if all of its rational points can be generated by a single point $P$, which is not torsion, together with some other points which are torsion. I don't know whether that's true in your case because I don't know what all the rational points are. You say you've found all the integer points, but you need to be able to generate all the rational points too.

(As a historical note, "torsion" means "twist." It's not at all obvious what the above definition has to do with twisting; this is actually quite a long story and involves algebraic topology, specifically some groups that appear in that subject which sometimes have nontrivial torsion in a way that is related to "twists" in some topological space. It would take quite awhile to explain what this all means but in any case it's not a random word that comes out of nowhere, there are specific historical reasons for it.)

If you haven't found it yet then I can warmly recommend Silverman and Tate's Rational Points on Elliptic Curves, which was written to be accessible to undergraduates and I think does a good job at conveying the essential ideas without getting bogged down in technical details.

Answer 2

I guess your curve is $y^2 = x^3 - 3x + 34$. The set of integral points on this curve is exactly the one you describe. I found that using the Sage Cell Server to make a short computation. Here is a link to a particular cell I made. It lists the points $(x,y)$ instead as $[x:y:z]$ as:

[(-3 : 4 : 1), (-1 : 6 : 1), (2 : 6 : 1), (5 : 12 : 1), (15 : 58 : 1), (29 : 156 : 1)]

The general way you find a curve $y^2 = x^3 + Ax + B$ on the Sage Cell Server is EllipticCurve([0,0,0,A,B]).

Let's talk about $P = (5,12)$. The answer is that $(5,12)$ is not a torsion a point. Here are the first few multiples of $P$: \begin{align*} P &=(5,12)\\ 2P &= (-1,6),\\ 3P &= (-3,-4),\\ 4P &= (2,-6),\\ 5P &= (29,-156),\\ 6P &= (15,58),\\ 7P &= (29/25,708/125),\\ 8P &= (-55/16, 123/64),\\ 9P &= (-11/81, -4276/729) \end{align*} and so on.

One definition of a torsion point is that the set $P$, $2P$, $3P$, ... eventually repeats. This condition recaptures the one that Qiaochu Yuan gave above, but it also provides a slightly different mode of thinking. To show $P$ is torsion, you just need to work you way back to $P$ by using $2P$, $3P$, and so on. All the points you travel through along the way will also be torsion points, by the way. Therefore, we use the indefinite article a torsion point rather than the definite the torsion point.

How do you prove a point $P$ is not torsion? Qiachou Yuan gave one possible response and you can happily use that. But you actually stumbled upon part of the calculation you can make as well, and I put it above on purpose. Once you reach the point $7P$ you find the fractional coordinates $$ 7P = (29/25, 708/125). $$ At that point, you can definitely say $P$ is not torsion. In fact, there is a theorem called the Nagell-Lutz Theorem that implies that the torsion points $Q$ are among the ones with whole number entries, which is the same as asking that $Q$, $2Q$, $3Q$, and so on all have whole number entries. The theorem holds for any elliptic curve $y^2 = x^3 + Ax + B$ where $A$ and $B$ are integers.

One benefit of the Nagell-Lutz criterion is that it requires less sophisticated technology to explain. It is in the second chapter of the Silverman-Tate reference Qiachou Yuan provided. I taught this once to a seminar course, and I remember the language of group theory being completely suppressed.

Nagell-Lutz is also the kind of criterion that you can see intuitively. The key point is that while torsion points cycle back to themselves, the other points have denominators that grow and grow. That's what you see with the $25$, $16$, $81$ in $7P$, $8P$, and $9P$ above. To take this more extreme, the $x$-coordinate of $50P = (x,y)$ has denominator: $$ 29325506545286020892676923992167095134964591049812393987794767409 $$ and that is nothing compared to $1000P$!

Finally, I address your second question. For each curve you have the torsion points (the ones that cycle back to themselves) and then you have the non-torsion points. The rank of the curve measures how many non-torsion points there are. For your curve $y^2 = x^3 - 3x + 34$ in fact you did stumble upon it having rank one, in the sense that every point is generated by your point $P$. You can find the numerical values like that using the Sage Cell Server or the LMFDB (linked in a comment above).

I'll have to leave it there today. Thank you for the interesting question.