Matt Satriano, Department of Pure Mathematics, University of Waterloo
"Short Ride in a Height Machine"
This talk is meant to be a quick introduction to the theory of height functions on projective varieties defined over number fields. After describing the fundamental case of heights on projective space, we go into the basics of Weil's height machine. For any projective variety, this gives a height function attached to each linear equivalence class of Cartier divisors (or each line bundle), satisfying a variety of pleasant functorial properties.
Alas, these height functions will only be defined up to bounded constants, so we spend the latter portion of the talk discussing canonical heights, a special choice of height function for which certain key properties hold without the pesky bounded constants. In particular, we single out two main examples where this is possible: projective space and abelian varieties. We conclude by tying canonical heights back to pre-periodic points of dynamical systems, deducing as a corollary that abelian varieties have finitely many torsion points.