The p-torsion subgroup scheme of an elliptic curve

TL;DR

This paper investigates which twisted forms of the p-torsion subgroup scheme of elliptic curves occur over various fields, revealing that not all forms appear in fields of characteristic p ≥ 13, impacting understanding of p-divisible groups.

Contribution

It demonstrates that most twisted forms of p occur as subgroup schemes of elliptic curves over certain fields, but not in all cases, especially for characteristic p  13 or higher, highlighting limitations in the occurrence.

Findings

Most twisted forms occur over finite and local fields for p  11.

Not all twisted forms occur in fields of characteristic p  13 or higher.

Implication that some p-divisible and formal groups do not arise from elliptic curves.

Abstract

Let $k$ be a field of positive characteristic $p$. Question: Does every twisted form of $\mu_p$ over $k$ occur as subgroup scheme of an elliptic curve over $k$? We show that this is true for most finite fields, for local fields and for fields of characteristic $p\leq11$. However, it is false in general for fields of characteristic $p\geq13$, which implies that there are also $p$-divisible and formal groups of height one over such fields that do not arise from elliptic curves. It also implies that the Hasse invariant does not obey the Hasse principle. Moreover, we also analyse twisted forms of $p$-torsion subgroup schemes of ordinary elliptic curves and the analogous questions for supersingular curves.