How arithmetic intersection theory on a proper flat model over Z turns the intuitive size of an algebraic solution into a number: the height of a point or subvariety as an arithmetic intersection degree of hermitian line bundles, then re-expressed through adelic metrics on Berkovich analytic spaces at every place. Students learn how semipositive and admissible metrics give canonical measures and arithmetic volumes, why a generic sequence of small points must equidistribute toward the canonical measure, and how that equidistribution forces a subvariety of an abelian variety to have height zero only when it is a torsion subvariety - the Bogomolov conjecture, with Manin-Mumford as a corollary.
Lecture notes from the 2017 Grenoble summer school (29 pages) that build the height machine of Diophantine geometry out of arithmetic intersection theory. A height measures the arithmetic size of an …