Torsors under some group schemes of order pn

AbstractLet R be a complete dvr with perfect residue field k of characteristic p>0. Let {Gλ}λ∈R be the class of R-affine, commutative, smooth of relative dimension one group schemes generically isomorphic to Gm. Let G:={Gλ,n}λ∈R,n∈N be the class of finite flat commutative group schemes of order pn defined as kernels of the isogenies φλ,n:Gλ→G(λpn). We provide an explicit description of torsors over R-schemes under the group schemes Gλ and Gλ,n