2024/02/13 by Maxim Mornev, Richard Pink, Mornev, Maxim +1
Mathematics · #Rings, Modules, and Algebras #Algebraic structures and combinatorial models #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.2402.08254
Let ϕ be a Drinfeld A-module of finite residual characteristic \mathfrakp over a local field K. We study the action of the inertia group of K on a modified adelic Tate module \smashT^∘ad(ϕ) which differs from the usual adelic Tate module only at the \mathfrakp-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