Hopf-Galois module structure of tame Cp x Cp

Let p be an odd prime number, K a number field containing a primitive pth root of unity, and L a Galois extension of K with Galois group isomorphic to C-p x C-p. We study in detail the local and global structure of the ring of integers D-L as a module over its associated order U-H in each of the Hopf algebras H giving nonclassical Hopf-Galois structures on the extension, complementing the p = 2 case considered in [12]. For each Hopf algebra giving a nonclassical Hopf-Galois structure on L/K we show that D-L is locally free over its associated order U-H in H, compute local generators, and determine necessary and sufficient conditions for D-L to be free over U-H.