  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.
