Some Hopf Galois structures arising from elementary abelian $p$-groups

Abstract:

Let $p$ be an odd prime, $G = Z_p^m$, the elementary abelian $p$-group of rank $m$, and let $\Gamma$ be the group of principal units of the ring $\mathbb {F}_p[x]/(x^{m+1})$. If $L/K$ is a Galois extension with Galois group $\Gamma$, then we show that for $p \ge 5$, the number of Hopf Galois structures on $L/K$ afforded by $K$-Hopf algebras with associated group $G$ is greater than $p^s$, where $s = \frac {(m-1)^2}3 - m$.