arXiv subjects
Publications and source records attributed to Laurent Berger.
The purpose of this note is to show that the computation of the universal norms in Iwasawa cohomology for the $p$-adic cyclotomic extension implies the same result for any $p$-adic extension that contains the cyclotomic extension. In addition, we prove a twisted cyclotomic variant, which applies to Lubin--Tate extensions.
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 $\mathbf{Q}_p$''.
The goal of this paper is to give a numerical criterion for an open question in $p$-adic Fourier theory. Let $F$ be a finite extension of $\mathbf{Q}_p$. Schneider and Teitelbaum defined and studied the character variety $\mathfrak{X}$, which is a rigid analytic curve over $F$ that parameterizes the set of locally $F$-analytic characters $λ: (o_F,+) \to (\mathbf{C}_p^\times,\times)$. Determining the structure of the ring $Λ_F(\mathfrak{X})$ of bounded-by-one functions on $\mathfrak{X}$ defined over $F$ seems like a difficult question. Using the Katz isomorphism, we prove that if $F= \mathbf{Q}_{p^2}$, then $Λ_F(\mathfrak{X}) = o_F [\![o_F]\!]$ if and only if the $o_F$-module of integer-valued polynomials on $o_F$ is generated by a certain explicit set. Some computations in SageMath indicate that this seems to be the case.
We show that the coefficients of a power series occurring in $p$-adic Fourier theory for $\mathbf{Q}_{p^2}$ have valuations that are given by an intriguing formula.
Fresnel and de Mathan proved that the p-adic Fourier transform is surjective. We reinterpret their result in terms of analytic boundaries, and extend it beyond the cyclotomic case. We also give some applications of their result to Schneider and Teitelbaum's p-adic Fourier theory, in particular to generalized Mahler expansions and to the geometry of the character variety.
Let $K$ be a $p$-adic field, and let $K_\infty/K$ be a Galois extension that is almost totally ramified, and whose Galois group is a $p$-adic Lie group of dimension $1$. We prove that $K_\infty$ is not dense in $(\mathbf{B}_{\mathrm{dR}}^+ / \operatorname{Fil}^2 \mathbf{B}_{\mathrm{dR}}^+ )^{\operatorname{Gal}(\overline{K}/K_\infty)}$. Moreover, the restriction of $θ$ to the closure of $K_\infty$ is injective, and its image via $θ$ is the set of vectors of $\widehat{K}_\infty$ that are $C^1$ with zero derivative for the action of $\operatorname{Gal}(K_\infty/K)$. The main ingredient for proving these results is the construction of an explicit lattice of $\mathcal{O}_{K_\infty}$ that is commensurable with $\mathcal{O}_{K_\infty}^{d=0}$, where $d : \mathcal{O}_{K_\infty} \to Ω_{\mathcal{O}_{K_\infty} / \mathcal{O}_K}$ is the differential.
Let E be a field of characteristic p. In a previous paper of ours, we defined and studied super-Hölder vectors in certain E-linear representations of Z_p. In the present paper, we define and study super-Hölder vectors in certain E-linear representations of a general p-adic Lie group. We then consider certain p-adic Lie extensions K_\infty / K of a p-adic field K, and compute the super-Hölder vectors in the tilt of K_\infty. We show that these super-Hölder vectors are the perfection of the field of norms of K_\infty / K. By specializing to the case of a Lubin-Tate extension, we are able to recover E((Y)) inside the Y-adic completion of its perfection, seen as a valued E-vector space endowed with the action of O_K^\times given by the endomorphisms of the corresponding Lubin-Tate group.
This paper is motivated by an open question in $p$-adic Fourier theory, that seems to be more difficult than it appears at first glance. Let $L$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $o_L$ and let $\mathbb{C}_p$ denote the completion of an algebraic closure of $\mathbb{Q}_p$. In their work on $p$-adic Fourier theory, Schneider and Teitelbaum defined and studied the character variety $\mathfrak{X}$. This character variety is a rigid analytic curve over $L$ that parameterizes the set of locally $L$-analytic characters $λ: (o_L,+) \to (\mathbb{C}_p^\times,\times)$. One of the main results of Schneider and Teitelbaum is that over $\mathbb{C}_p$, the curve $\mathfrak{X}$ becomes isomorphic to the open unit disk. Let $Λ_L(\mathfrak{X})$ denote the ring of bounded-by-one functions on $\mathfrak{X}$. If $μ\in o_L [\![o_L]\!]$ is a measure on $o_L$, then $λ\mapsto μ(λ)$ gives rise to an element of $Λ_L(\mathfrak{X})$. The resulting map $o_L [\![o_L]\!] \to Λ_L(\mathfrak{X})$ is injective. The question is: do we have $Λ_L(\mathfrak{X}) = o_L [\![o_L]\!]$? In this paper, we prove various results that were obtained while studying this question. In particular, we give several criteria for a positive answer to the above question. We also recall and prove the ``Katz isomorphism'' that describes the dual of a certain space of continuous functions on $o_L$. An important part of our paper is devoted to providing a proof of this theorem which was stated in 1977 by Katz. We then show how it applies to the question. Besides $p$-adic Fourier theory, the above question is related to the theory of formal groups, the theory of integer valued polynomials on $o_L$, $p$-adic Hodge theory, and Iwasawa theory.
Let $E$ be a field of characteristic $p$. The group $\mathbf{Z}_p^\times$ acts on $E((X))$ by $a \cdot f(X) = f((1+X)^a-1)$. This action extends to the $X$-adic completion $\tilde{\mathbf{E}}$ of $\cup_{n \geq 0} E((X^{1/p^n}))$. We show how to recover $E((X))$ from the valued $E$-vector space $\tilde{\mathbf{E}}$ endowed with its action of $\mathbf{Z}_p^\times$. To do this, we introduce the notion of super-Hölder vector in certain $E$-linear representations of $\mathbf{Z}_p$. This is a characteristic $p$ analogue of the notion of locally analytic vector in $p$-adic Banach representations of $p$-adic Lie groups.
Let LT be a Lubin-Tate formal group attached to a finite extension of Qp. By a theorem of Lubin-Sarkis, an invertible characteristic p power series that commutes with the elements of Aut(LT) is itself in Aut(LT). We extend this result to perfectoid power series, by lifting such a power series to characteristic zero and using the theory of locally analytic vectors in certain rings of p-adic periods. This allows us to recover the field of norms of the Lubin-Tate extension from its completed perfection.
We ask several questions about substitution maps in the Robba ring. These questions are motivated by $p$-adic Hodge theory and the theory of $p$-adic dynamical systems. We provide answers to those questions in special cases, thereby generalizing results of Kedlaya, Colmez, and others.
We show that if V and V' are two p-adic representations of Gal(Qp^alg/Qp) whose tensor product is trianguline, then V and V' are both potentially trianguline.
We define the resultant of two power series with coefficients in the ring of integers of a $p$-adic field. In order to do this, we prove a universal version of the Weierstrass preparation theorem.
Let K be a p-adic field and let F and G be two formal groups over O_K. We prove that if F and G have infinitely many torsion points in common, then F=G. This follows from a rigidity result: any bounded power series that sends infinitely many torsion points of F to torsion points of F is an endomorphism of F.
We prove two theorems that confirm an observation of Lubin concerning families of $p$-adic power series that commute under composition: under certain conditions, there is a formal group such that the power series in the family are either endomorphisms of this group, or semi-conjugate to endomorphisms of this group.
Let $K$ be a finite extension of $\mathbf{Q}_p$. We use the theory of $(φ,Γ)$-modules in the Lubin-Tate setting to construct some corestriction-compatible families of classes in the cohomology of $V$, for certain representations $V$ of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/K)$. If in addition $V$ is crystalline, we describe these classes explicitly using Bloch-Kato's exponential maps. This allows us to generalize Perrin-Riou's period map to the Lubin-Tate setting.
We give a short proof of a conjecture of Lubin concerning certain families of $p$-adic power series that commute under composition. We prove that if the family is full (large enough), there exists a Lubin-Tate formal group such that all the power series in the family are endomorphisms of this group. The proof uses ramification theory and some $p$-adic Hodge theory.
Let $K$ be a finite extension of $\mathbf{Q}_p$ and let $G_K = \mathrm{Gal}(\bar{\mathbf{Q}}_p/K)$. There is a very useful classification of $p$-adic representations of $G_K$ in terms of cyclotomic $(φ,Γ)$-modules (cyclotomic means that $Γ={\rm Gal}(K_\infty/K)$ where $K_\infty$ is the cyclotomic extension of $K$). One particularly convenient feature of the cyclotomic theory is the fact that any $(φ,Γ)$-module is overconvergent. Questions pertaining to the $p$-adic local Langlands correspondence lead us to ask for a generalization of the theory of $(φ,Γ)$-modules, with the cyclotomic extension replaced by an infinitely ramified $p$-adic Lie extension $K_\infty / K$. It is not clear what shape such a generalization should have in general. Even in the case where we have such a generalization, namely the case of a Lubin-Tate extension, most $(φ,Γ)$-modules fail to be overconvergent. In this article, we develop an approach that gives a solution to both problems at the same time, by considering the locally analytic vectors for the action of $Γ$ inside some big modules defined using Fontaine's rings of periods. We show that, in the cyclotomic case, we recover the ususal overconvergent $(φ,Γ)$-modules. In the Lubin-Tate case, we can prove, as an application of our theory, a folklore conjecture in the field stating that $(φ,Γ)$-modules attached to $F$-analytic representations are overconvergent.