Locally analytic vectors and $\mathbf{Z}_p$-extensions

TL;DR

The paper analyzes locally analytic vectors in the integral, mixed-characteristic framework for ${\mathbf Z}_p$-extensions replacing the cyclotomic tower, using integral locally analytic notions (super-Hölder) to study potential liftings of the field of norms and the resulting ${\varphi,\Gamma}$-module theories. It establishes a precise equivalence for ${K_\infty/K}$: the existence of nontrivial locally analytic vectors in ${\widetilde{\mathbf{A}}}_K^{\dagger}$ is equivalent to the existence of an overconvergent lift of the field of norms; in the unramified base case, nontrivial LA vectors force the extension to be twisted cyclotomic up to finite extension. Consequently, Kedlaya's conjecture fails for many ${\mathbf Z}_p$-extensions not of twisted cyclotomic type (notably the anticyclotomic one over ${\mathbf Q}_{p^2}$), while the analysis reveals higher locally analytic vectors in some settings. The work blends Lubin-Tate theory, field-of-norms techniques, and p-adic Hodge theory to classify LA vectors and their implications for integral $(\varphi,\Gamma)$-module theory in Lubin-Tate-type towers.

Abstract

Let $K$ be a finite extension of $\mathbf{Q}_p$ and let $\mathcal{G}_K = \mathrm{Gal}(\overline{\mathbf{Q}_p}/K)$. Lately, interest has risen around a generalization of the theory of $(\varphi,Γ)$-modules, replacing the cyclotomic extension with an arbitrary infinitely ramified $p$-adic Lie extension. Computations from Berger suggest that locally analytic vectors should provide such a generalization for any arbitrary infinitely ramified $p$-adic Lie extension, and this has been conjectured by Kedlaya. In this paper, we focus on the case of $\mathbf{Z}_p$-extensions, using recent work of Berger-Rozensztajn and Porat on an integral version of locally analytic vectors and explain what can be the structure of the locally analytic vectors in the higher rings of periods $\widetilde{\mathbf{A}}^{\dagger}$ in this setting. We show that the existence of nontrivial locally analytic vectors in $\widetilde{\mathbf{A}}^{\dagger}$, a necessary condition for Kedlaya's conjecture to hold, is equivalent to the existence of an overconvergent lift of the field of norms attached to the $\mathbf{Z}_p$-extension. Finally, in the case where $K/\mathbf{Q}_p$ is unramified, we are able to prove that the only extensions for which such nontrivial locally analytic vectors exist are exactly the twisted cyclotomic extensions, up to a finite extension. In particular, this disproves Kedlaya's conjecture and also shows that there is no overconvergent lift of the field of norms in the anticyclotomic setting.

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