Torsors under some group schemes of order [formula omitted

Abstract

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