### Canonical subgroups for p-divisible groups

- Series
- Algebra Seminar
- Time
- Thursday, March 17, 2011 - 16:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Joe Rabinoff – Harvard University

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.