Integral points on an elliptic curve

Solution 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.

Solution 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.

Solution 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.