Integral points on an elliptic curve

Question

Let's start with an elliptic curve in the form $$E : y^2 = x^3 + Ax + B, \qquad A, B \in \mathbb{Z}.$$ I am wondering about integral points. I know that Siegel proved that $E$ has only finitely many integral points. I know Nagell and Lutz proved that every non $\mathcal{O}$ torsion point has coordinates that are integers.

Can any of you tell me anything else we would know? Here are a few questions I have come up with but if there is any other interesting thing to say I would love to know it. Answers to any of these would be great.

1. Can we say anything interesting about non-torsion integral points? (I don't have an idea of what "interesting" means exactly, maybe they are in a sort form or related to torsion points somehow)
2. Are there bounds for the number of or biggest integral points?
3. Do people keep track of records for most or biggest integral points?
4. If so, any idea of what these records are?
5. Anything else?

Thanks

Answer 1

As you know, your curve may have infinitely many rational points. Now suppose it has a rational point $(r/t,s/t)$, so $s^2/t^2=r^3/t^3+Ar/t+B$, so $(st^2)^2=(rt)^3+At^4(rt)+Bt^6$, so $(rt,st)$ is an integral point on the elliptic curve $y^2=x^3+A'x+B'$. It follows that there is no finite bound to the number of integral points on an elliptic curve. It suggests that the bigger the coefficients, the more integral points are possible.

A related question that may interest you is that of the rank of your curve, the number of independent generators of the group of rational points. It is believed, but, I think, not proved, that the rank is unbounded, but it's hard to find examples with large rank (where large might mean more than 20), and people do go to some effort to set new records. According to Wikipedia, curves with rank at least 28 are known.

Answer 2

There are bounds for the size of the integral points on an elliptic curve over $\mathbb{Q}$. For example there's Baker's famous result that if $A, B, C, D \in \mathbb{Z}$ are such that $\max{ \{|A|, |B|, |C|, |D| \} } \leq H$ then any integral point $(x, y) \in E(\mathbb{Q})$ satisfies

$$\max{ \{ |x|, |y| \} } < e^{(10^6 H)^{10^6}}$$

where $E: Y^2 = AX^3 + BX^2 + CX + D$, this is quoted from Theorem 5.4 in Chapter IX of Silverman's book The Arithmetic of Elliptic Curves.

Also from Silverman's book, there's conjecture 7.4 in that same chapter which says the following.

(Hall-Lang Conjecture) There exists constants $C$ and $r$ such that for every elliptic curve $E/\mathbb{Q}$ with Weierstrass equation

$$Y^2 = X^3 + AX + B$$

where $A, B \in \mathbb{Z}$, and for every integral point $(x, y) \in \mathbb{Z}^2$ with $(x, y) \in E(\mathbb{Q})$ the following inequality holds

$$|x| \leq C (\max{ \{ |A|, |B| \} })^r$$

You can find lots of really interesting information in Chapter IX of Silverman's book.

Answer 3

The most important point to note about integral points is that the set of integral points is not a property of the elliptic curve, but rather of the specific Weierstrass model. If $(x,y)=(n,m)$ is an integral solution to $y^2=x^3+Ax+B$, then for any $t\in \mathbb{N}$, the substitution $(x',y')=(t^2x,t^3y)$ gives a new Weierstrass model of the same elliptic curve, but this new model will have the integral point $(t^2n,t^3m)$. So the hunt for records is not only non-sensical if you allow the curve to vary, it doesn't even make sense on a fixed elliptic curve.

What is a worthwhile research question is to find optimal bounds for the heights of integral solutions on a given Weierstrass equation. That is still an active area of research. If you have that, then you can start with generators of the Mordell-Weil group and keep multiplying them until you exceed the bound. This gives an effective method to find all integral solutions (provided you could find the generators of the Mordell-Weil group). Sometimes, much more elementary methods also get you there, see e.g. Adrián's and my cooperative answer to a similar question. Such elementary methods motivate the study of ideal class groups and units in rings of integers of number fields.