# Why are integer points on elliptic curves interesting and useful?

I read some papers which dealed with integer points on elliptic curves. One of these papers are

My question is: Why are integer points on elliptic curves useful? One thing I could imagine:

If there is a method which tells you that a give set of integer points:

$P_1,\dots,P_l$ is independent (in terms of generation). So this would give a lower bound for the rank.

Is this possible? Even for special cases?

If not, is there anything else useful?

• I'm not sure why there are votes to close. Admittedly the question could be better phrased to be clearer, but especially the part asking about independence of integer points in the group of rational points is very interesting. Nov 7, 2015 at 18:57
• I believe it's unknown whether there's an absolute bound for the number of integer points on the global minimal model of an elliptic curve over the rationals (as we range over all elliptic curves). That such a natural Diophantine question is still open is for me a fine reason to think about integer points on elliptic curves.
– eric
Nov 7, 2015 at 22:57

To answer the second part of your question. there is a conjecture of Serge Lang which says that the number of integer points (on a minimal Weierstrass equation) is bounded solely in terms of the rank. The conjecture is known to be true if one assumes the $ABC$-conjecture, in the stronger form $$\#E(\mathbb Z) \le C^{1+\operatorname{rank}E(\mathbb Q)}$$ for an absolute constant $C$. So a sequence of curves with $\#E_i(\mathbb Z)\to\infty$ would (conjecturally) give a sequence with unbounded rank. However, there are those who believe that there should be an absolute upper bound for $\#E(\mathbb Z)$, and recently some who believe that there should be an absolute upper bound for $\operatorname{rank}E(\mathbb Q)$. Anyway, lots of interesting open questions.
For the first part of your question, there are many natural problems that end up coming down to integer points on elliptic curves. Just as there are many natural problems that come down to rational points on curves of genus $g\ge2$. But I'm not sure that's the right answer to your question. For me, it comes down to (1) integers and their relationships to one another are interesting, (2) the basic operations on the integers are addition and multiplication, so the basic functions are polynomial functions, (3) hence Diophantine equations (polynomial equations to be solved in integers) are intrinsically interesting. If you'll grant that, then people eventually realized that the "right" way to classify equations of the form $f(x,y)=0$ was by genus, the first case ($g=0$) is now well understood, and the second case ($g=1$) still presents challenges.
• Thanks a lot for this nice answer! What about the independence of integer points? I think there must be a way to show that two Points $P_1$ and $P_2$ are independent. For example you can find several rank record on this side: [this] side. I noticed that many generating points are integer points, so it seems the authors used some technique to show that the points are independent. : web.math.pmf.unizg.hr/~duje/tors/rankhist.html Nov 9, 2015 at 17:30
• @Benjamin The technique being used to generate high rank also tends to produce lots of integer points. I would guess that they then just start taking subsets and computing the canonical height pairing matrices to find a set that is independent. My recollection is that Elkies rank 28 curve has well over 100 points with integer coords, so they're not all independent. Another favorite example of mine is $y^2=x^3+17$, which has rank 2 and has 8 pairs of integer points. Nov 9, 2015 at 20:30