Γ(p)-Level Structure on p-Divisible Groups

Abstract

The main result of the thesis is the introduction of a notion of Γ(p)-level structure for p-divisible groups. This generalizes the Drinfeld-Katz-Mazur notion of full level structure for 1-dimensional p-divisible groups. The associated moduli problem has a natural forgetful map to the Γ0(p)-level moduli problem. Exploiting this map and known results about Γ0(p)-level, we show that our notion yields a flat moduli problem. We show that in the case of 1-dimensional p-divisible groups, it coincides with the existing Drinfeld-Katz-Mazur notion. In the second half of the thesis, we introduce a notion of epipelagic level structure. As part of the task of writing down a local model for the associated moduli problem, one needs to understand commutative finite flat group schemes G of order p2 killed by p, equipped with an extension structures 0→ H1→ G→ H2→ 0, where H1,H2 are finite flat of order p. We investigate a particular class of extensions, namely extensions of Z/pZ by μp over Zp-algebras. These can be classified using Kummer theory. We present a different approach, which leads to a more explicit classification.