A p-adic family of dihedral (phi,Gamma)-modules

Relative p-adic Hodge theory, II: (phi, Gamma)-modules

Building on foundations introduced in a previous paper, we give several p-adic analytic descriptions of the categories of etale Zp-local systems and etale Qp-local systems on an affinoid algebra over a finite extension of Qp (or more generally, over the fraction field of the Witt vectors of a perfect field of characteristic p). These include generalizations of Fontaine's theory of (phi, Gamma)-modules, the refinement of Fontaine's construction introduced by Cherbonnier and Colmez, and a recent description by Fargues and Fontaine in terms of semistable vector bundles on a certain scheme. Our de…

New methods for (phi, Gamma)-modules

We provide new proofs of two key results of p-adic Hodge theory: the Fontaine-Wintenberger isomorphism between Galois groups in characteristic 0 and characteristic p, and the Cherbonnier-Colmez theorem on decompletion of (phi, Gamma)-modules. These proofs are derived from joint work with Liu on relative p-adic Hodge theory, and are closely related to Scholze's study of perfectoid algebras and spaces.

What is worthy of investigation? Philosophical attitudes and their impact on mathematical development by the example of discovering 10-adic numbers

We describe in dialogue form a possible way of discovering and investigating 10-adic numbers starting from the naive question about a `largest natural number'. Among the topics we pursue are possibilities of extensions to transfinite 10-adic numbers, 10-adic representations of rational numbers, zero divisors, square roots and 10-adic roots of higher degree of natural numbers, and applications of 10-adic number representation in computer arithmetic. The participants of the dialogue are idealized embodiments of different philosophical attitudes towards mathematics. The article aims at illustrati…

Cohomology and Duality for (phi,Gamma)-modules over the Robba ring

Given a p-adic representation of the Galois group of a local field, we show that its Galois cohomology can be computed using the associated etale (phi,Gamma)-module over the Robba ring; this is a variant of a result of Herr. We then establish analogues, for not necessarily etale (phi,Gamma)-modules over the Robba ring, of the Euler-Poincare characteristic formula and Tate local duality for p-adic representations. These results are expected to intervene in the duality theory for Selmer groups associated to de Rham representations.

Rankin-Selberg Euler systems and p-adic interpolation

We construct motivic cohomology classes attached to Rankin--Selberg convolutions of modular forms of weights $\ge 2$, show that these vary analytically in p-adic families, and relate their image under the p-adic regulator map to values of L-functions. As consequences, we prove new cases of Perrin-Riou's conjecture on motivic L-values; we prove finiteness results for Tate--Shafarevich groups for twists of elliptic curves by dihedral Artin characters; and we prove one inclusion in the Iwasawa main conjecture for a single modular form over an imaginary quadratic field.

Local theta-regulators of an algebraic number -- p-adic Conjectures

Let K/Q be Galois and let eta in K* be such that the multiplicative Z[G]-module generated by eta is of Z-rank n.We define the local theta-regulators Delta\_p^theta(eta) in F\_p for the Q\_p-irreducible characters theta of G=Gal(K/Q). Let V\_theta be the theta-irreducible representation. A linear representation L^theta=delta.V\_theta is associated withDelta\_p^theta(eta) whose nullity is equivalent to delta$\ge$1 (Theorem 3.9). Each Delta\_p^theta(eta) yields Reg\_p^theta(eta) modulo p in the factorization $\prod$\_theta (Reg\_p^theta(eta))^phi(1) of Reg\_p^G(eta) := Reg\_p(eta)/p^[K : Q] (norm…

On arithmetic families of filtered phi-modules and crystalline representations

We consider stacks of filtered phi-modules over rigid analytic spaces and adic spaces. We show that these modules parametrize p-adic Galois representations of the absolute Galois group of a p-adic field with varying coefficients over an open substack containing all classical points. Further we study a period morphism (defined by Pappas and Rapoport) from a stack parametrizing integral data and determine the image of this morphism.

Représentations galoisiennes p-adiques et (phi,tau)-modules

Let p be an odd prime number and K be a p-adic field. In this paper, we develop an analogue of Fontaine's theory of (phi,Gamma)-modules replacing the p-cyclotomic extension by the extension K_infty obtained by adding to K a compatible system of p^n-th roots of a fixed uniformizer pi of K. As a result, we obtain a new classification of p-adic representations of G_K = Gal(Kbar/K) by some (phi, τ)-modules. We then make a link between the theory of (phi,tau)-modules discussed above and the so-called theory of (phi,N_nabla)$-modules developped by Kisin. As a corollary, we answer a question of Tong …

Relative p-adic Hodge theory, II: Imperfect period rings

In a previous paper, we constructed a category of (phi, Gamma)-modules associated to any adic space over Q_p with the property that the etale (phi, Gamma)-modules correspond to etale Q_p-local systems; these involve sheaves of period rings for Scholze's pro-etale topology. In this paper, we first extend Kiehl's theory of coherent sheaves on rigid analytic spaces to a theory of pseudocoherent sheaves on adic spaces, then construct a corresponding theory of pseudocoherent (phi, Gamma)-modules. We then relate these objects to a more explicit construction in case the space comes equipped with a su…

On Families of (Phi,Gamma)-modules

Berger and Colmez introduced a theory of families of overconvergent étale (Phi,Gamma)-modules associated to families of p-adic Galois representations over p-adic Banach algebras. However, in contrast with the classical theory of (Phi,Gamma)-modules, the functor they obtain is not an equivalence of categories. In this paper, we prove that when the base is an affinoid space, every family of (overconvergent) étale (Phi,Gamma)-modules can locally be converted into a family of p-adic representations in a unique manner, providing the "local" equivalence. There is a global mod p obstruction related t…

Semisimple types for p-adic classical groups

We construct, for any symplectic, unitary or special orthogonal group over a locally compact nonarchimedean local field of odd residual characteristic, a type for each Bernstein component of the category of smooth representations, using Bushnell--Kutzko's theory of covers. Moreover, for a component corresponding to a cuspidal representation of a maximal Levi subgroup, we prove that the Hecke algebra is either abelian, or a generic Hecke algebra on an infinite dihedral group, with parameters which are, at least in principle, computable via results of Lusztig.

A generalization of Kato's local epsilon-conjecture for (phi,Gamma)-modules over the Robba ring

The aim of this article is to generalize Kato's (commutative) p-adic local epsilon-conjecture [Ka93b] for families of (phi,Gamma)-modules over the Robba ring. In particular, we prove the generalized local epsilon-conjecture for rank one (phi,Gamma)-modules, which is a generalization of Kato's theorem [Ka93b] for rank one Galois representations. The key ingredients are the recent results of Kedlaya-Pottharst-Xiao [KPX12] on the finiteness of cohomology of (phi,Gamma)-modules and the theory of Bloch-Kato's exponential map for (phi,Gamma)-modules developed in [Na13].

Sur la classification de quelques phi-modules simples

This note is an appendix to a preprint by E. Hellmann. We give a complete classification of simple objects of the category of vector spaces D over K = Fpbar((u)) equipped with an endomorphism phi whose image generates D and that are semi-linear with respect to the ring morphism sending u to u^b (b > 1 is an integer) and acting on elements of k through a fixed automorphism. Some of these phi-modules are involved in the classification of finite flat group schemes over ring of integers of p-adic fields.

Local epsilon conjecture and p-adic differential equations

Laurent Berger attached a p-adic differential equation N_rig(M) with a Frobenius structure to an arbitrary de Rham (phi, Gamma)-module over a Robba ring. In this article, we compare the local epsilon conjecture for the cyclotomic deformation of M with that of N_rig(M). We first define an isomorphism between the fundamental lines of their cyclotomic deformations using the second author's results on the big exponential map. As a main result of the article, we show that this isomorphism enables us to reduce the local epsilon conjecture for the cyclotomic deformation of M to that of N_rig(M).

Frobenius modules over multivariate Robba rings

We describe a class of multivariate series rings generalizing the usual Robba ring over a p-adic field, and give a basic development of phi-modules over such rings. This makes it possible to give a unified survey of a number of recent developments in p-adic Hodge theory.

广义phi-模的模堆

Moduli stacks of generalized phi-modules

该研究改进了Emerton-Gee堆的群变换代表性结论，展示了其在p进域上的有限呈现代数堆结构。

p-adic interpolation of automorphic periods for GL(2)

We give a new and representation theoretic construction of $p$-adic interpolation series for central values of self-dual Rankin-Selberg $L$-functions for $\operatorname{GL}_2$ in dihedral towers of CM fields, using expressions of these central values as automorphic periods. The main novelty of this construction, apart from the level of generality in which it works, is that it is completely local. We give the construction here for a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real field corresponding to a $\mathfrak{p}$-ordinary Hilbert modular forms of parallel …

Sur la densité des représentations cristallines du groupe de Galois absolu de Q_p

Let X_d be the p-adic analytic space classifying the d-dimensional (semisimple) p-adic Galois representations of the absolute Galois group of Q_p. We show that the crystalline representations are Zarski-dense in many irreducible components of 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 Gouvêa-Mazur in this context, based on a study of analytic families of trianguline (phi,Gamma)-modules over the Robba ring. We show in partic…

GL₂(Qₚ)上的范畴p进Langlands对应

A categorical p-adic Langlands correspondence for GL_2(Q_p)

本文构建了一个从GL₂(Qₚ)的p进表示导出范畴到(phi,Gamma)-模堆上诱导凝聚sheaves导出范畴的全忠实函子，揭示了p进Langlands对应中的范畴结构。

The p-adic monodromy theorem in the imperfect residue field case

Let K be a complete discrete valuation field of mixed characteristic (0,p) and G_K the absolute Galois group of K. In this paper, we will prove the p-adic monodromy theorem for p-adic representations of G_K without any assumption on the residue field of K, for example the finiteness of a p-basis of the residue field of K. The main point of the proof is a construction of (phi,G_K)-module Nrig^+(V) for a de Rham representation V, which is a generalization of Pierre Colmez' Nrig^+(V). In particular, our proof is essentially different from Kazuma Morita's proof in the case when the residue field a…

Moduli stacks of étale (phi,Gamma)-modules and the existence of crystalline lifts

We construct stacks which algebraize Mazur's formal deformation rings of local Galois representations. More precisely, we construct Noetherian formal algebraic stacks over Spf Zp which parameterize étale (phi,Gamma)-modules; the formal completions of these stacks at points in their special fibres recover the universal deformation rings of local Galois representations. We use these stacks to show that all mod p representations of the absolute Galois group of a p-adic local field lift to characteristic zero, and indeed admit crystalline lifts. We also discuss the relationship between the geometr…

平衡三重积$p$进$L$函数与Stark点

Balanced triple product $p$-adic $L$-functions and Stark points

本研究推广Hsieh的三变量平衡三重积$p$-进$L$-函数构造，针对秩一情形提出联系其插值域外值与Stark点、Stark单位的椭圆Stark猜想，并对同一虚二次域的二面体表示证明该猜想。

Global applications of relative (phi-Gamma)-modules I

In this paper, given a smooth proper scheme X over a p-adic dvr and a p-power torsion etale local system L on it, we study a family of sheaves associated to the cohomology of local relative (Phi-Gamma)-modules of L and their cohomology. As applications we derive descriptions of the etale cohomology groups on the geometric generic fiber of X with values in L, as well as of their classical (Phi-Gamma)-modules, in terms of cohomology of the above mentioned sheaves.

Invitation to higher local fields, Part II, section 6: Phi-Gamma-modules and Galois cohomology

This is is a survey of applications of Fontaine's theory of p-adic representations of local fields (Phi-Gamma-modules) to Galois cohomology of local fields and explicit formulas for the Hilbert symbol in relation with two-dimensional local objects.

B-pairs and (phi,Gamma)-modules

Main change from v1 : theorem C has been modified, see remark 3.1.7 (2). 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.

Trianguline representations

Trianguline representations are a certain class of p-adic representations of Gal(Qp^alg/Qp) like the crystalline, semistable and de Rham representations of Fontaine. Their definition involves the theory of (phi,Gamma)-modules. In this survey, we explain the theory of (phi,Gamma)-modules and the definition and properties of trianguline representations. After that, we give some examples of their occurrence in arithmetic geometry.

Introduction to moduli, l-adic representations and the Regular Version of the Inverse Galois Problem

Sect 1 introduces Nielsen classes attached to (G,C), where C is r conjugacy classes in a finite group G, and a braid action on them. These give reduced Hurwitz spaces, denoted H(G,C)^rd. The section concludes with a braid formula for the genus of these spaces when r = 4. If there is at least one prime l for which G is divisible by l, but has no Z/l quotient, then there is a canonical tower of reduced Hurwitz spaces over H(G,C)^rd, using the Universal Frattini cover, ~G, of G, and ~G_ab, its abelianized version. The towers are nonempty assuming C are l' classes satisfying a cohomological condit…

Sur quelques représentations potentiellement cristallines de GL_2(Q_p)

To each 2-dimensional irreducible p-adic representation of Gal(Qpbar/Qp) which becomes crystalline over an abelian extension of Q_p, we associate a Banach space B(V) endowed with a linear continuous unitary action of GL_2(Q_p). When V is moreover phi-semi-simple, we use the (phi,Gamma)-module and the Wach module associated to V to show that the representation B(V) is nonzero, topologically irreducible and admissible.

De Rham representations and universal norms

We compute the module of universal norms for a de Rham p-adic representation. The computation uses the theory of (phi,Gamma)-modules (Cherbonnier-Colmez's reciprocity formula) and the differential equation attached to a de Rham representation.

On categories of (phi, Gamma)-modules

Let K be a complete discretely valued field of mixed characteristics (0, p) with perfect residue field. One of the central objects of study in p-adic Hodge theory is the category of continuous representations of the absolute Galois group of K on finite-dimensional Qp-vector spaces. In recent years, it has become clear that this category can be studied more effectively by embedding it into the larger category of (phi, Gamma)-modules; this larger category plays a role analogous to that played by the category of vector bundles on a compact Riemann surface in the Narasimhan-Seshadri theorem on uni…

Wach modules, regulator maps, and epsilon-isomorphisms in families

We prove the local epsilon-isomorphism conjecture of Fukaya and Kato [FK06] for certain crystalline families of G_Qp-representations. This conjecture can be regarded as a local analogue of the Iwasawa main conjecture for families. Our work extends earlier work of Kato for rank-1 modules (cf. [Ven13]), of Benois and Berger for crystalline G_Qp-representations with respect to the cyclotomic extension (cf. [BB08]), as well as of Loeffler, Venjakob, and Zerbes (cf. [LVZ13]) for crystalline G_Qp- representations with respect to abelian p-adic Lie extensions of Qp. Nakamura [Nak13, Nak14] has also f…

Surfaces and p-adic fields I: Dehn twists

Following the philosophy of arithmetic topology, we describe a point of view which helps look at surfaces and $p$-adic fields in a "uniform way", and show that results on mapping class groups can be extended to this point of view, and thus be applied to $G_{K}$, the absolute Galois groups of the $p$-adic field $K$. By moving both groups to the world of pro-$p$ groups (for $G_{K}$ we take its maximal pro-$p$ quotient, and for $π_{1}(S)$ we take its pro-$p$ completion), we see they both are pro-$p$ Poincare duality groups of dimension $2$, also known as Demuskin groups. Such groups have a very n…