arXiv subjects

Léo Poyeton

Publications and source records attributed to Léo Poyeton.

10 recordsLinked to original sources

arXiv2026

de Rham theory and locally analytic vectors

Let $K_\infty/K$ be a $p$-adic Lie extension of a $p$-adic field $K$. We study the subring of pro-analytic vectors in the de Rham period ring $\mathbf{B}_{\mathrm{dR}}^+(K_\infty)$. We show that the pro-analytic subring admits a Galois-equivariant isomorphism with a formal power series ring $\widehat{K}_{\infty}^{\mathrm{la}} [[t_{K_\infty}]]$ if and only if $K_\infty$ satisfies a certain orientability condition, which says that the $\widehat{K}_\infty$-level Sen operator admits a Galois-equivariant $\mathbf{B}_{\mathrm{dR}}^+$-lift. A key input is the vanishing of higher locally analytic vectors of $\widehat{K}_\infty$-representations. As an application, we show that the lifted Sen operator induces regular connections on pro-analytic vectors of $\mathbf{B}_{\mathrm{dR}}^+$-representations, and can be used to compute Galois cohomology.

math.NT↗

arXiv2026

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

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 $(φ,Γ)$-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. In the anticyclotomic setting, assuming that such an overconvergent lift exists, we are able to construct elements in the corresponding Robba ring which should not exist according to a conjecture of Berger. We then prove that in this specific setting, a particular case of Berger's conjecture holds, discarding the existence of such elements. In particular, this disproves Kedlaya's conjecture and shows that there is no overconvergent lift of the field of norms in the anticyclotomic setting.

math.NT↗

arXiv2025

Multivariable $p$-adic Hodge theory for products of Galois groups

In this paper we explain how to attach to a family of $p$-adic representations of a product of Galois groups an overconvergent family of multivariable $(φ,Γ)$-modules, generalizing results from Pal-Zabradi and Carter-Kedlaya-Zabradi, using Colmez-Sen-Tate descent. We also define rings of multivariable crystalline and semistable periods, and explain how to recover this multivariable $p$-adic theory attached to a family of representations from its multivariable $(φ,Γ)$-module. We also explain how our framework allows us to recover the main results of Brinon-Chiarellotto-Mazzari on multivariable $p$-adic Galois representations.

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arXiv2024

Locally analytic vectors and rings of periods

In this paper, we try to extend Berger's and Colmez's point of view, using locally analytic vectors in order to generalize classical cyclotomic theory, in higher rings of periods. We also explain how the formalism of locally analytic vectors recovers the ring $\mathbf{B}_{Sen}$ of Colmez, and extends to Sen theory in the de Rham case, and to classical $(φ,Γ)$-modules theory. We explain what happens when we try to generalize constructions of $(φ,Γ)$-modules to arbitrary infinitely ramified $p$-adic Lie extensions, and provide a conjecture on the structure of the locally analytic vectors in the corresponding rings. We also highlight the fact that the situation should be very different, depending on wether the $p$-adic Lie extension ``contains a cyclotomic extension'' or not. Finally, we explain how some of these constructions may be related to the construction of a ring of trianguline periods.

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arXiv2023

A criterion for Lubin's conjecture

We prove that a formulation of a conjecture of Lubin regarding two power series commuting for the composition is equivalent to a criterion of checking that some extensions generated by the nonarchimedean dynamical system arising from the power series are Galois. As a consequence of this criterion, we obtain a proof of Lubin's conjecture in a new case.

math.NT↗

arXiv2022

F-analytic B-pairs

In this note, we define the notion of $F$-analytic $B$-pairs and we prove that its category is equivalent to the one of $F$-analytic $(φ_q,Γ_K)$-modules.

math.NT↗

arXiv2021

Families of Galois representations and $(φ, τ)$-modules

Let $p$ be a prime, and let $K$ be a finite extension of $\mathbf{Q}_p$, with absolute Galois group $\cal{G}_K$. Let $π$ be a uniformizer of $K$ and let $K_\infty$ be the Kummer extension obtained by adjoining to $K$ a system of compatible $p^n$-th roots of $π$, for all $n$, and let $L$ be the Galois closure of $K_\infty$. Using these extensions, Caruso has constructed étale $(φ,τ)$-modules, which classify $p$-adic Galois representations of $K$. In this paper, we use locally analytic vectors and theories of families of $φ$-modules over Robba rings to prove the overconvergence of $(φ,τ)$-modules in families. As examples, we also compute some explicit families of $(φ,τ)$-modules in some simple cases.

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arXiv2019

$(ϕ,τ)$-modules différentiels et représentations potentiellement semi-stables

Soit $K$ un corps $p$-adique et soit $V$ une représentation $p$-adique de $\mathcal{G}_K = \mathrm{Gal}(\bar{K}/K)$. La surconvergence des $(ϕ,τ)$-modules nous permet d'attacher à $V$ un $ϕ$-module différentiel à connexion $D_{τ,\mathrm{rig}}^\dagger(V)$ sur l'anneau de Robba $\mathbf{B}_{τ,\mathrm{rig},K}^\dagger$. On montre dans cet article comment retrouver les invariants $D_{\mathrm{cris}}(V)$ et $D_{\mathrm{st}}(V)$ à partir de $D_{τ,\mathrm{rig}}^\dagger(V)$, et comment caractériser les représentations potentiellement semi-stables, ainsi que celles de $E$-hauteur finie, à partir de la connexion. Let $K$ be a $p$-adic field and let $V$ be a $p$-adic representation of $\mathcal{G}_K=\mathrm{Gal}(\bar{K}/K)$. The overconvergence of $(ϕ,τ)$-modules allows us to attach to $V$ a differential $ϕ$-module $D_{τ,\mathrm{rig}}^\dagger(V)$ on the Robba ring $\mathbf{B}_{τ,\mathrm{rig},K}^\dagger$ that comes equipped with a connection. We show in this paper how to recover the invariants $D_{\mathrm{cris}}(V)$ and $D_{\mathrm{st}}(V)$ from $D_{τ,\mathrm{rig}}^\dagger(V)$, and give a characterization of both potentially semi-stable representations of $\mathcal{G}_K$ and finite $E$-height representations in terms of the connection operator.

math.NT↗

arXiv2019

Formal groups and lifts of the field of norms

Let $K$ be a finite extension of $\mathbf{Q}_p$. The field of norms of a strictly APF extension $K_\infty/K$ is a local field of characteristic $p$ equipped with an action of $\mathrm{Gal}(K_\infty/K)$. When can we lift this action to characteristic zero, along with a compatible Frobenius map ? In this article, we explain what we mean by lifting the field of norms, explain its relevance to the theory of $(φ,Γ)$-modules, and show that under a certain assumption on the type of lift, such an extension is generated by the torsion points of a relative Lubin-Tate group and that the power series giving the lift of the action of the Galois group of $K_\infty/K$ are twists of semi-conjugates of endomorphisms of the same relative Lubin-Tate group.

math.NT↗

arXiv2019

Locally analytic vectors and overconvergent $(φ, τ)$-modules

Let $p$ be a prime, let $K$ be a complete discrete valuation field of characteristic $0$ with a perfect residue field of characteristic $p$, and let $G_K$ be the Galois group. Let $π$ be a fixed uniformizer of $K$, let $K_\infty$ be the extension by adjoining to $K$ a system of compatible $p^n$-th roots of $π$ for all $n$, and let $L$ be the Galois closure of $K_\infty$. Using these field extensions, Caruso constructs the $(φ, τ)$-modules, which classify $p$-adic Galois representations of $G_K$. In this paper, we study locally analytic vectors in some period rings with respect to the $p$-adic Lie group $\mathrm{Gal}(L/K)$, in the spirit of the work by Berger and Colmez. Using these locally analytic vectors, and using the classical overconvergent $(φ, Γ)$-modules, we can establish the overconvergence property of the $(φ, τ)$-modules.

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