Uniformity of stably integral points

Abstract. The uniformity conjecture (attributed to Mazur) predicts that the number of rational point in curves of genus > 1 are not only finite (Faltings’s Theorem) but uniform; in particular their number depends only on the genus and on the base field. In 1997 it has been proved that this follows from Lang Conjecture in arbitrary dimension. The key point is a “fibered power theorem” which is a purely geometric result on fiber powers of fibered families. In this talk we will introduce the known results, deepening the interplay of geometry and arithmetic, as well as presenting new results for uniformity of (stably) integral points for curves and a fibered power theorem for pairs. This is joint work with Kenneth Ascher (Brown University).