Georgia Institute of Technology College of Sciences

An elliptic curve over the integer ring of a p-adic field whose special fiber is ordinary has a canonical line contained in its p-torsion. This fact has many arithmetic applications: for instance, it shows that there is a canonical partially-defined section of the natural map of modular curves X_0(Np) -> X_0(N). Lubin was the first to notice that elliptic curves with "not too supersingular" reduction also contain a canonical order-p subgroup. I'll begin the talk by giving an overview of Lubin and Katz's theory of the canonical subgroup of an elliptic curve. I'll then explain one approach to defining the canonical subgroup of any abelian variety (even any p-divisible group), and state a very general existence result. If there is time I'll indicate the role tropical geometry plays in its proof.