Endomorphism rings of formal 𝐴₀-modules

Abstract

Let A 0 {A_0} be the valuation ring of a finite extension K 0 {K_0} of Q p {Q_p} and A ⊃ A 0 A \supset {A_0} be a complete discrete valuation ring with the perfect residue field. We consider the endomorphism rings of n n -dimensional formal A 0 {A_0} -modules Γ \Gamma over A A of finite A 0 {A_0} -height with reduction absolutely simple up to isogeny. Especially we prove commutativity of End A , A 0 ( Γ ) {\operatorname {End} _{A,{A_0}}}(\Gamma ) . Given an arbitrary finite unramified extension K 1 {K_1} of K 0 {K_0} , a variety of examples (different dimensions and different A 0 {A_0} -heights) is constructed whose absolute endomorphism rings are isomorphic to the valuation ring of K 1 {K_1} .