Integral points on elliptic curves and 3-torsion in class groups

We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques and methods based on quasi-orthogonality in the Mordell-Weil lattice. We apply our results to break previous bounds on the number of elliptic curves of given conductor and the size of the 3 3 -torsion part of the class group of a quadratic field. The same ideas can be used to count rational points on curves of higher genus.