Robba Ring Research Articles

Substitution maps in the Robba ring

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.

A group-theoretic generalization of thep-adic local monodromy theorem

AbstractLetGbe a connected reductive group over ap-adic number fieldF. We propose and study the notions ofG-$\varphi $-modules andG-$(\varphi ,\nabla )$-modules over the Robba ring, which are exact faithfulF-linear tensor functors from the category ofG-representations on finite-dimensionalF-vector spaces to the categories of$\varphi $-modules and$(\varphi ,\nabla )$-modules over the Robba ring, respectively, commuting with the respective fiber functors. We study Kedlaya’s slope filtration theorem in this context, and show thatG-$(\varphi ,\nabla )$-modules over the Robba ring are “G-quasi-unipotent,” which is a generalization of thep-adic local monodromy theorem proved independently by Y. André, K. S. Kedlaya, and Z. Mebkhout.

Logarithmic growth filtrations for -modules over the bounded Robba ring

In the 1970s, Dwork defined the logarithmic growth (log-growth for short) filtrations for $p$-adic differential equations $Dx=0$ on the $p$-adic open unit disc $|t|<1$, which measure the asymptotic behavior of solutions $x$ as $|t|\to 1^{-}$. Then, Dwork calculated the log-growth filtration for $p$-adic Gaussian hypergeometric differential equation. In the late 2000s, Chiarellotto and Tsuzuki proposed a fundamental conjecture on the log-growth filtrations for $(\varphi ,\nabla )$-modules over $K[\![t]\!]_0$, which can be regarded as a generalization of Dwork's calculation. In this paper, we prove a generalization of the conjecture to $(\varphi ,\nabla )$-modules over the bounded Robba ring. As an application, we prove a generalization of Dwork's conjecture proposed by Chiarellotto and Tsuzuki on the specialization property for log-growth Newton polygons.

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

Let K be a p-adic field and let V be a p-adic representation of G K =Gal(K ¯/K). The overconvergence of (ϕ,τ)-modules allows us to attach to V a differential ϕ-module D τ,rig † (V) on the Robba ring B τ,rig,K † that comes equipped with a connection. We show in this paper how to recover the invariants D cris (V) and D st (V) from D τ,rig † (V), and give a characterization of both potentially semi-stable representations of 𝒢 K and finite E-height representations in terms of the connection operator.

Arithmetic Families of $(\varphi,\Gamma)$-Modules and Locally Analytic Representations of $GL_2(Q_p)$

Let A be a \bold{Q}_p -affinoid algebra, in the sense of Tate. We develop a theory of locally convex A -modules parallel to the treatment in the case of a field by Schneider and Teitelbaum. We prove that there is an integration map linking a category of locally analytic representations in A -modules and separately continuous relative distribution modules. There is a suitable theory of locally analytic cohomology for these objects and a version of Shapiro's Lemma, generalizing results of Kohlhaase. As an application we propose a p -adic Langlands correspondence in families: For a regular trianguline (\varphi,\Gamma) -module of dimension 2 over the relative Robba ring \mathcal R_A we construct a locally analytic GL_2(Q_p) -representation in A -modules.

Iterated line integrals over Laurent series fields of characteristic p

Inspired by Besser’s work on Coleman integration, we use ∇-modules to define iterated line integrals over Laurent series fields of characteristic p taking values in double cosets of unipotent n×n matrices with coefficients in the Robba ring divided out by unipotent n×n matrices with coefficients in the bounded Robba ring on the left and by unipotent n×n matrices with coefficients in the constant field on the right. We reach our definition by looking at the analogous theory for Laurent series fields of characteristic 0 first, and reinterpreting the classical formal logarithm in terms of ∇-modules on formal schemes. To illustrate that the new p-adic theory is non-trivial, we show that it includes the p-adic formal logarithm as a special case.

Wach Modules, Regulator Maps, and $\varepsilon$-Isomorphisms in Families

Abstract We prove the “local $\varepsilon$-isomorphism” conjecture of Fukaya and Kato [13] for certain crystalline families of $G_{{\mathbf{Q}_p}}$-representations. This conjecture can be regarded as a local analog of the Iwasawa main conjecture for families. Our work extends earlier work of Kato for rank-$1$ modules (cf. [33]), of Benois and Berger for crystalline $G_{{\mathbf{Q}_p}}$-representations with respect to the cyclotomic extension (cf. [1]), as well as of Loeffler et al. (cf. [21]) for crystalline $G_{{\mathbf{Q}_p}}$-representations with respect to abelian $p$-adic Lie extensions of ${\mathbf{Q}_p}$. Nakamura [24, 25] has also formulated a version of Kato’s $\varepsilon$-conjecture for affinoid families of $(\varphi,\Gamma)$-modules over the Robba ring, and proved his conjecture in the rank-$1$ case. He used this case to construct an $\varepsilon$-isomorphism for families of trianguline $(\varphi,\Gamma)$-modules, depending on a fixed triangulation. Our results imply that this $\varepsilon$-isomorphism is independent of the chosen triangulation for certain crystalline families. The main ingredient of our proof consists of the construction of families of Wach modules generalizing work of Wach and Berger [6] and following Kisin’s approach to the construction of potentially semi-stable deformation rings [18].

Fundamental groups and good reduction criteria for curves over positive characteristic local fields

In this article I define and study the overconvergent rigid fundamental group of a variety over an equicharacteristic local field. This is a non-abelian $(\varphi,\nabla)$-module over the bounded Robba ring $\mathcal{E}_K^\dagger$, whose underlying unipotent group (after base changing to the Amice ring $\mathcal{E}_K$) is exactly the classical rigid fundamental group. I then use this to prove an equicharacteristic, $p$-adic analogue of Oda's theorem that a semistable curve over a $p$-adic field has good reduction iff the Galois action on its $\ell$-adic unipotent fundamental group is unramified.

The logarithmic growth of element of Robba ring which satisfies Frobenius equation over bounded Robba ring

We study the logarithmic growth of an element of the Robba ring which satisfies a Frobenius equation over the bounded Robba ring. Chiarellotto and Tsuzuki computed the logarithmic growth of analytic functions on the open unit disc with coefficients in a $p$-adic local field which satisfy Frobenius equations over bounded functions of rank 2. We extend their result by replacing those functions by elements of the Robba ring which satisfy Frobenius equations over the bounded Robba ring. Moreover, we will see, in special cases, the zeros of these functions have some cyclicity and the logarithmic growth can be computed by the zeros of these function.

Sur la densité des représentations cristallines de $$\text{ Gal}(\overline{\mathbb Q }_p/\mathbb Q _p)$$

Let \({\mathfrak X }_d\) be the \(p\)-adic analytic space classifying the semisimple continuous representations \(\text{ Gal}(\overline{\mathbb Q }_p/\mathbb Q _p) \rightarrow \mathrm GL _d(\overline{\mathbb Q }_p)\). We show that the crystalline representations are Zarski-dense in many irreducible components of \({\mathfrak X }_d\), including the components made of residually irreducible representations. This extends to any dimension \(d\) previous results of Colmez and Kisin for \(d = 2\). For this we construct an analogue of the infinite fern of Gouvea–Mazur in this context, based on a study of analytic families of trianguline \((\varphi ,\Gamma )\)-modules over the Robba ring. We show in particular the existence of a universal family of (framed, regular) trianguline \((\varphi ,\Gamma )\)-modules, as well as the density of the crystalline \((\varphi ,\Gamma )\)-modules in this family. These results may be viewed as a local analogue of the theory of \(p\)-adic families of finite slope automorphic forms and they are new already in dimension \(2\). The technical heart of the paper is a collection of results about the Fontaine–Herr cohomology of families of trianguline \((\varphi ,\Gamma )\)-modules.

Slope filtrations in families

This paper concerns arithmetic families of -modules over reduced affinoid spaces. For such a family, we first prove that the slope polygons are lower semicontinuous around any rigid point. We further prove that if the slope polygons are locally constant around a rigid point, then around this point, the family has a global slope filtration after base change to some extended Robba ring.

Generalized Robba rings

We prove that any projective coadmissible module over the locally analytic distribution algebra of a compact $p$-adic Lie group is finitely generated. In particular, the category of coadmissible modules does not have enough projectives. In the Appendix a "generalized Robba ring" for uniform pro-$p$ groups is constructed which naturally contains the locally analytic distribution algebra as a subring. The construction uses the theory of generalized microlocalization of quasi-abelian normed algebras that is also developed there. We equip this generalized Robba ring with a self-dual locally convex topology extending the topology on the distribution algebra. This is used to show some results on coadmissible modules.

Classification of two-dimensional split trianguline representations ofp-adic fields

AbstractThe aim of this article is to classify two-dimensional split trianguline representations ofp-adic fields. This is a generalization of a result of Colmez who classified two-dimensional split trianguline representations of$\mathrm {Gal}(\bar {\mathbb {Q}}_p/\mathbb {Q}_p)$forp≠2 by using (φ,Γ)-modules over a Robba ring. In this article, for any primepand for anyp-adic fieldK, we classify two-dimensional split trianguline representations of$\mathrm {Gal}(\bar {K}/K)$usingB-pairs as defined by Berger.

Construction de (φ,Γ)-modules: représentationsp-adiques etB-paires

We study the category of B-pairs (W_e,W_dR^+) where W_e is a free B_cris^{phi=1}-module with a semilinear and continuous action of G_K and where W_dR^+ is a G_K-stable B_dR^+ -lattice in B_dR \otimes W_e. This category contains the category of p-adic representations and is naturally equivalent to the category of all (phi,Gamma)-modules over the Robba ring

Rank one solvable p-adic differential equations and finite Abelian characters via Lubin–Tate groups

We introduce a new class of exponentials of Artin–Hasse type, called π-exponentials. These exponentials depend on the choice of a generator π of the Tate module of a Lubin–Tate group \(\mathfrak{G}\) over \(\mathbb{Z}_p\). They arise naturally as solutions of solvable differential modules over the Robba ring. If \(\mathfrak{G}\) is isomorphic to \(\widehat{\mathbb{G}}_m\) over \(\mathbb{Z}_p\), we develop methods to test their over-convergence, and get in this way a stronger version of the Frobenius structure theorem for differential equations. We define a natural transformation of the Artin–Schreier complex into the Kummer complex. This provides an explicit generator of the Kummer unramified extension of \(\epsilon^\dagger_{K_{\infty}}\), whose residue field is a given Artin–Schreier extension of \(k(\!(t)\!)\), where k is the residue field of K. We then compute explicitly the group, under tensor product, of isomorphism classes of rank one solvable differential equations. Moreover, we get a canonical way to compute the rank one φ-module over \(\epsilon^\dagger_{K_\infty}\) attached to a rank one representation of \(\mathrm{Gal}(k(\!(t)\!)^{\mathrm{sep}}/k(\!(t)\!))\), defined by an Artin–Schreier character.

Slope filtrations revisited

We give a “second generation” exposition of the slope filtration theorem for modules with Frobenius action over the Robba ring, providing a number of simplifications in the arguments. Some of these are inspired by parallel work of Hartl and Pink, which points out some analogies with the formalism of stable vector bundles.

Représentations p -adiques et équations différentielles

In this paper, we associate to every $p$-adic representation $V$ a $p$-adic differential equation $\mathbf{D}^{\dagger}_{\mathrm{rig}}(V)$, that is to say a module with a connection over the Robba ring. We do this via the theory of Fontaine's $(\phi,\Gamma_K)$-modules. This construction enables us to relate the theory of $(\phi,\Gamma_K)$-modules to $p$-adic Hodge theory. We explain how to construct $mathbf{D}_{mathrm{cris}}(V)$ and $\mathbf{D}_{\mathrm{st}}(V)$ from $\mathbf{D}^{\dagger}_{\mathrm{rig}}(V)$, which allows us to recognize semi-stable or crystalline representations; the connection is then either unipotent or trivial on $\mathbf{D}^{\dagger}_{\mathrm{rig}}(V)[1/t]$. In general, the connection has an infinite number of regular singularities, but we show that $V$ is de Rham if and only if those are apparent singularities. A structure theorem for modules over the Robba ring allows us to get rid of all singularities at once, and to obtain a ``classical'' differential equation, with a Frobenius structure. A recent theorem of Y. Andr\'e gives a complete description of the structure of such an object. This allows us to prove Fontaine's $p$-adic monodromy conjecture: every de Rham representation is potentially semi-stable. As an application, we can extend to the case of arbitrary perfect residue fields some results of Hyodo ($H^1_g=H^1_{st}$), of Perrin-Riou (the semi-stability of ordinary representations), of Colmez (absolutely crystalline representations are of finite height), and of Bloch and Kato (if the weights of $V$ are $\geq 2$, then Bloch-Kato's exponential $\exp_V$ is an isomorphism).