Galois endomorphisms of the torsion subgroup of one-parameter generic formal groups

Abstract:

Let ${{\mathbf {Z}}_p}$ be the ring of $p$-adic integers and let $\Gamma$ be a one-parameter generic formal group of finite height $h$ defined over ${{\mathbf {Z}}_p}\left [ {\left [ {{t_1}, \ldots ,{t_{h - 1}}} \right ]} \right ] = A$. Let $K$ be the field of fractions of $A,G = \operatorname {Gal}\left ( {\bar K/K} \right )$ and $T\left ( \Gamma \right )$ the Tate module of $\Gamma$. The purpose of this paper is to give an elementary proof that the $\operatorname {map}\operatorname {End}_{A}\left ( \Gamma \right ) \to \operatorname {End}_{G}\left ( {T\left ( \Gamma \right )} \right )$ is a surjection.