[thaumasiotes]

> Well, thinking abstractly, if there are only finitely many rational solutions, then there certainly exists a second equation, g(x,y) = 0, giving another curve C' that intersects C at only the rational points. (Because any finite set of points can be interpolated by a curve, e.g. by Newton interpolation. [shrug] Nothing deep about this!)

Ok, any finite set of points can be interpolated by a curve.

Why is it obvious that one of these interpolated curves will necessarily avoid intersecting C at any other point?

[jlev1]

Good point, it's not necessarily possible! That said, it would be enough to know that the intersection just contains all the rational points (and is finite, though that's automatic for 2+ polynomials in two variables with no common factors). Then we can just check them one by one, discarding the non-rational points.

Alternately, it's possible the construction gives a system of auxiliary equations, which, together with f(x,y) = 0, pick out the rational points of the curve. (The term "variety", as in "Selmer variety", means solution set to a system of polynomial equations). Still, short of knowing the points in advance, I wouldn't know how to easily produce such equations.

[im3w1l]

I can think of counterexamples if we allow C to self-intersect: We can then make a loop and pick one point outside it and one point inside and zero on it.