Integer points belonging to two distinct elliptic curves.

Question

Two different circles can have an integer point in common (for example, $P=(1,1)$ belongs to both $x^2+y^2-2=0$ and $x^2+y^2-4(x+y)+6=0$) but any pair of distinct elliptic curves on the class defined over $\mathbb {Q}$ by $X^3+Y^3=A$ where $A$ is a cube-free integer, excepting for the trivial $0=(1,-1)$, can not. One wonders on a family of elliptic curves defined over $\mathbb {Q}$ in which there are couples of them that -as with the circles- may have an integer point in common. How to express this family as with the class $X^3+Y^3=A$, I mean analitically?

Answer 1

I'm not quite sure what you mean. Given two curves $f(X,Y) = 0$ and $g(X,Y) = 0$ where $f$ and $g$ are polynomials, let $r_X(Y)$ and $r_Y(X)$ be the resultants of $f(X,Y)$ and $g(X,Y)$ with respect to $X$ and $Y$ respectively. A point $(x,y)$ common to both curves must satisfy $r_X(y) = r_Y(x) = 0$. In particular, if you want a common integer solution, a necessary condition is that $r_X$ and $r_Y$ must both have integer roots.

EDIT (incorporating last comment): If you want two continuous parametric families of curves that always intersect in at least one integer point, then by continuity that point is constant. Choose the intersection point, and construct the curves to pass through that point. Thus the curves $Y^2+X^3+aX+b=0$ and $X^2+Y^3+cY+d=0$ intersect at $(1,1)$ if $a+b+2=c+d+2=0$. In almost all cases, these are elliptic.