Full Level Structure on Some Group Schemes

TL;DR

This paper defines a concept of full level structure on specific group schemes and demonstrates the absence of a natural notion of such structures over the entire stack of finite flat commutative group schemes.

Contribution

It introduces a new definition of full level structure for certain group schemes and clarifies its limitations over the broader class of all finite flat commutative group schemes.

Findings

No natural notion of full level structure exists over the stack of all finite flat commutative group schemes.

The paper provides a specific definition for group schemes of the form G×G where G has rank p.

The work highlights structural limitations in extending level structures universally.

Abstract

We give a definition of full level structure on group schemes of the form $G\times G$, where $G$ is a finite flat commutative group scheme of rank $p$ over a $\mathbb{Z}_p$-scheme $S$ or, more generally, a truncated $p$-divisible group of height $1$. We show that there is no natural notion of full level structure over the stack of all finite flat commutative group schemes.