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p-DIVISIBLE GROUPS : PART II

This talk discusses the main results of Tate’s paper "p-Divisible Groups" [6]. From the point of view of p-adic Hodge theory, this is a foundational paper and within this setting, much of the technical work being done becomes extremely important. From our perspective, having just learned what p-divisible groups are three weeks ago, however, the significance of the results could easily be drowned in details. I will thus proceed in an unorthodox fashion beginning with the end result and relegating a lot of technical details to appendices and references. Before we begin, I mention a rather amazing motivation for our study of p-divisible groups. In Faltings’ proof and in many other instances, one is interested in studying the structure of moduli spaces of abelian varieties (of fixed dimension, polarization of fixed degree, etc.). The primary tool for studying the local structure of moduli spaces is deformation theory. Deformation theory tells us information about the completed stalks of the structure sheaf on our moduli space. A natural question one might ask is given an abelian variety A over Fp, can we lift it to an abelian scheme over Zp? For example, in the case of elliptic curves, all we have to do is lift the Weierstrass equation and so there are many possible lifts. Given that lifts exist, we could try to lift an abelian variety together with some endomorphisms or maybe two abelian varieties with homomorphisms between them. The first step would be to pose the same problem for Zp/pZp as opposed to Zp. This is an infinitessimal deformation problem from characteristic p, and a remarkable theorem of Serre-Tate says that in residue characteristic p, the deformation theory of an abelian variety is the same as that of its p-divisible group: