Local Kummer theory for Drinfeld modules

Let φ be a Drinfeld A-module of finite residual characteristic p̄ over a local field K. We study the action of the inertia group of K on a modified adelic Tate module T^∘ad(φ) which differs from the usual adelic Tate module only at the p̄-primary component. After replacing K by a finite extension we can assume that φ is the analytic quotient of a Drinfeld module ψ of good reduction by a lattice M⊂ K. The image of inertia acting on T^∘ad(φ) is then naturally a subgroup of HomA(M,T^∘ₐd(ψ)). This subgroup is described by a canonical local Kummer pairing that we study extensively in this article. In particular we give an effective formula for the image of inertia up to finite index, and obtain a necessary and sufficient condition for this image to be open. We also determine the image of the ramification filtration