Integral points in families of elliptic curves

Abstract:

Taking a family of elliptic curves and imposing some ordering, we may ask how often a curve has an integral point and how many integral points there are on average. We expect that elliptic curves with any nontrivial integral points are generally very sparse. In certain quadratic and cubic twist families, it is possible to show that almost all curves contain no nontrivial integral points. The proof uses the parameterisation of integral points by integral binary forms and the distribution of Selmer groups.