On Abelian Hopf Galois structures and finite commutative nilpotent rings

Let G be an elementary abelian p-group of rank n, with p an odd prime. In order to count the Hopf Galois structures of type G on a Galois extension of fields with Galois group G, we need to determine the orbits under conjugation by Aut(G) of regular subgroups of the holomorph of G that are isomorphic to G. The orbits correspond to isomorphism types of commutative nilpotent Fp-algebras N of dimension n with N = 0. Adapting arguments of Kruse and Price, we obtain lower and upper bounds on the number fc(n) of isomorphism types of commutative nilpotent algebras N of dimension n (as vector spaces) over the field Fp satisfying N = 0. For n = 3, 4 there are five, resp. eleven isomorphism types of commutative nilpotent algebras, independent of p (for p > 3). For n ≥ 6, we show that fc(n) depends on p. In particular, for n = 6 we show that fc(n) ≥ b(p− 1)/6c by adapting an argument of Suprunenko and Tyschkevich. For n ≥ 7, fc(n) ≥ pn−6. Conjecturally, fc(5) is finite and independent of p, but that case remains open. Finally, applying a result of Poonen, we observe that the number of Hopf Galois structures of type G is asymptotic to fc(n) as n goes to infinity.