Galois measures and the Katz map
Laurent Berger
TL;DR
The paper addresses the problem of describing Galois measures for Tate modules of height $2$ formal groups over the ring of integers of a finite unramified extension of $\mathbf{Q}_p$. It builds a framework around the covariant bialgebra $U(G)$ and its $p$-adic completion $\widehat{U}(G)$, via the Katz map $\mathcal{K}: \widehat{U}(G) \to C^0_{\operatorname{Gal}}(T_pH, \mathcal{O}_{\mathbf{C}_p})$ with $\mathcal{K}(u)(t)=\langle u, t(X)\rangle$, and analyzes its dual $\mathcal{K}^*$ and a $\psi$-integrality condition. The main contributions prove Katz's injectivity and characterize the image $\mathcal{K}^*(\mu)$ of Galois measures in terms of $\psi$-integrality (Theorem A), and, in this work, establish Theorem B: $\mathcal{K}$ is injective and for any $t\in T_p^\times H$, the map $\mathcal{K}_t$ is surjective onto $\mathcal{O}_{K_\infty}$, with $K_\infty$ the field generated by the $t_n$. The results extend prior Lubin–Tate cases to all height-two formal groups and illuminate a perfectoid-like structure in $\widehat{U}(G)$, connecting p-adic Hodge theory and Galois representations via Ax–Sen–Tate-type phenomena. This work solidifies the Katz framework for Galois measures in the height-two setting and provides tools potentially useful for non-archimedean harmonic analysis and Iwasawa-theoretic studies of formal groups.
Abstract
The purpose of this paper is to explain the proofs of the results announced by Nick Katz in 1977, namely a description of ``Galois measures for Tate modules of height two formal groups over the ring of integers of a finite unramified extension of $\mathbf{Q}_p$''.