p-jets Of p-isogenies

TL;DR

This paper investigates the behavior of p-jets of p-isogenies, revealing that while they are often not finite or flat, certain structures like p-divisible groups exhibit more regular behavior in the limit, especially for elliptic curves.

Contribution

The paper demonstrates how p-jets of p-isogenies behave differently for p-divisible groups, particularly in the context of elliptic curves and formal groups, highlighting the limit behavior mod p.

Findings

p-jets of p-isogenies are generally not finite or flat

p-divisible groups show improved structure in the limit mod p

The structure for ordinary elliptic curves depends on the Serre-Tate parameter

Abstract

p-jets of finite flat maps of schemes are generally neither finite nor flat. This phenomenon can be seen already in the case of p-isogenies of group schemes. However, for p-divisible groups, this pathology tends to disappear mod p "in the limit". We illustrate this in the case of the p-divisible groups of elliptic curves and of formal groups of finite height; in the case of ordinary elliptic curves the structure in the limit depends on the Serre-Tate parameter.