turkmath.org

It is a theorem of Siegel that the Weierstrass model $y^2 = x^3 + Ax + B$ of an elliptic curve has finitely many integral points. A ''random'' such curve should have no points at all. I will show that the average number of integral points on such curves (ordered by height) is bounded - in fact, by 66. The methods combine a Mumford-type gap principle, LP bounds in sphere packing, and results in Diophantine approximation. The same result also holds (though I have not computed an explicit constant) for the families \[y^2 = x^3 + Ax, y^2 = x^3 + B, \textrm{ and } y^2 = x^3 - n^2 x.\] If I have time I will also mention why the average is strictly smaller than one assuming the minimalist conjecture (that 50% of curves have rank zero and 50% have rank one).