The p-torsion subgroup scheme of an elliptic curve

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.