Leonidas Alaoglu Memorial Lecture in Mathematics

Fermat's Last Theorem, Faltings' theorems and recent results of Bhargava and others guarantee that certain large classes of algebraic curves have 'few' rational points. Heuristics - some of them developed very recently - suggest the rarity of rational points, or the rarity of 'new' rational points on a curve when one passes to a larger number field. In this lecture I will discuss this general subject and introduce joint work with Karl Rubin - also in the context of large fields of algebraic numbers.