Number Theory Seminar

Farbod ShokriehCornell University
Heights and tropical geometry

Friday, November 10, 2017 - 2:30pm
Malott 206

Given a polarized abelian variety A over a number field (or a function field), one can naturally extract two real numbers that capture the "complexity" of A: one is the Faltings height and the other is the Néron-Tate height (of a symmetric effective divisor defining the polarization). I will discuss a precise conjecture on the relationship between these two numbers, relating them to some subtle invariants arising from tropical geometry (more precisely, from Berkovich analytic spaces). I will then discuss the case of Jacobians, where the conjecture is now a theorem, and the proof uses combinatorics, convex geometry, and electrical networks. (Ongoing joint work with Robin de Jong.)