Elementary abelian Hopf Galois structures and polynomial formal groups

We find a relationship between regular embeddings of G, an elementary abelian p-group of order pn, into unipotent upper triangular matrices with entries in Fp and commutative dimension n degree 2 polynomial formal groups with nilpotent upper triangular structure matrices. We classify the latter when n=3 up to linear isomorphism, and use that classification to determine the number of Hopf Galois structures on a Galois extension L/K of fields with Galois group G elementary abelian of order p3. We also obtain a lower bound on the number of Hopf Galois structures on a Galois extension L/K when the Galois group is elementary abelian of order pn.