Hopf orders in $K[C_p^3]$ in characteristic $p$

Abstract

Let $p$ be prime, let $K$ be a non-archimedean local field of characteristic $p$ and let $C_p^3$ denote the elementary abelian group of order $p^3$. We give a complete classification of Hopf orders in the $K$-Hopf algebra $(K[C_p^3]{)}^{\ast }$ and under a mild condition compute their dual Hopf orders in the group ring $K[C_p^3]$.