The Zilber–Pink conjecture unifies various Diophantine questions—including the Mordell, Manin–Mumford, and Mordell–Lang conjectures in the context of abelian varieties, and the André–Oort conjecture in the context of Shimura varieties, which are driven by the principle that “Geometry governs Arithmetic”. In the particular case of curves, these conjectures reduce to finiteness statements about “special points” lying on a curve, which I will recall first.

Then I will discuss a $p$-adic method combining Buium’s theory of arithmetic jets ($p$-jets) and Coleman’s theory of $p$-adic integration to study the Zilber–Pink conjecture. For curves inside abelian varieties, this method provides an explicit bound under favourable conditions. This talk is based on recent and ongoing projects with Netan Dogra (King’s College London).