A conjecture of Breuil, Buzzard, and Emerton says that the slopes of certain reducible $p$-adic Galois representations must be integers. In previous work we showed this conjecture for representations that lie over certain non-subtle components of weight space. This article is a continuation of that work in which we completely classify the aforementioned representations over the non-subtle components of weight space, both for integer and non-integer slopes.
We compute reduction $\bar ρ$ of 3-dimensional irreducible crystalline representations $ρ$ of $G_{\mathbb Q_p}$ with Hodge-Tate weights $\{0, r , s\}$ satisfying $2 \leq r \leq p-2, \ \ 2+p \leq s \leq r + p-2.$ If $\bar ρ$ is irreducible then we show that the generic fiber of crystalline universal deformation ring of $\bar ρ$ is connected.
We complete the calculations begun in [BG09], using the p-adic local Langlands correspondence for GL2(Q_p) to give a complete description of the reduction modulo p of the 2-dimensional crystalline representations of G_{Q_p} of slope less than 1, when p > 2.
We use the p-adic local Langlands correspondence for GL_2(Q_p) to explicitly compute the reduction modulo p of crystalline representations of small slope, and give applications to modular forms.
The question of computing the reductions modulo $p$ of two-dimensional crystalline $p$-adic Galois representations has been studied extensively, and partial progress has been made for representations that have small weights, very small slopes, or very large slopes. It was conjectured by Breuil, Buzzard, and Emerton that these reductions are irreducible if they have even weight and non-integer slope. We prove some instances of this conjecture for slopes up to $\frac{p-1}{2}$.
We study the mod $p$ reduction of crystalline local Galois representations of dimension 2 under certain conditions on its weight and slope. Berger showed that for a fixed non-zero trace of the Frobenius, the reduction process is locally constant for varying weights. By explicit computation we obtain an upper bound that is a linear function of the slope, for the radius of this local constancy around some special points in the weight space.
Fontaine's $D_{\mathrm{cris}}$ functor allow us to associate an isocrystal to any crystalline representation. For a reductive group $G$, we study the reduction of lattices inside a germ of crystalline representations with $G$-structure $V$ to lattices (which are crystals) with $G$-structure inside $D_{\mathrm{cris}}(V)$. Using Kisin modules theory, we give a description of this reduction in terms of $G$, in the case when the representation $V$ is ($G$-)ordinary. In order to do that, first we need to generalize Fargues' construction of the Harder-Narasimhan filtration for $p$-divisible groups t…
We use the p-adic local Langlands correspondence for GL_2(Q_p) to find the reduction modulo p of certain two-dimensional crystalline Galois representations. In particular, we resolve a conjecture of Breuil, Buzzard, and Emerton in the case when the slope is strictly between one and three, and prove partial results towards this conjecture for arbitrary slopes. Moreover, we partially classify the reduction modulo p of these representations when the slope is equal to one.
We determine rational Kisin modules associated with two-dimensional, irreducible, crystalline representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ of Hodge-Tate weights $0, k-1$. If the slope is larger than $\lfloor \frac{k-1}{p} \rfloor$, we further identify an integral Kisin module, which we use to calculate the semisimple reduction of the Galois representation. In that range, we find that the reduction is constant, thereby improving on a theorem of Berger, Li, and Zhu.
We describe an algorithm to compute the reduction modulo $p$ of a crystalline Galois representation of dimension $2$ of $\text{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$ with distinct Hodge-Tate weights via the semi-simple modulo $p$ Langlands correspondence. We give some examples computed with an implementation of this algorithm in SAGE.
Let K_{f} be the finite unramified extension of Q_{p} of degree f and E any finite large enough coefficient field containing K_{f}. We construct analytic families of étale (Phi,Gamma)-modules which give rise to families of crystalline E-representations of the absolute Galois group G_{K_{f}} of K_{f}. For any irreducible effective two-dimensional crystalline E-representation of G_{K_{f}} with labeled Hodge-Tate weights {0,-k_{i}}_{τ_{i}} induced from a crystalline character of G_{K_{2f}}, we construct an infinite family of crystalline E-representations of G_{K_{f}} of the same Hodge-Tate type w…
Let $p$ be an odd prime number, $K_{f}$ the finite unramified extension of $\mathbb{Q}_{p}$ of degree $f$, and $G_{K_{f}}$ its absolute Galois group. We construct analytic families of étale $(φ,Γ)$-modules which give rise to some families of 2-dimensional crystalline representations of $G_{K_{f}}$ with length of filtration $\geq p$. As an application, we prove that the modulo $p$ reductions of the members of each such family (with respect to appropriately chosen Galois-stable lattices) are constant.
We prove the existence of local constancy phenomena for reductions in a general prime power setting of two-dimensional irreducible crystalline representations. Up to twist, these representations depend on two parameters: a trace $a_p$ and a weight $k$. We find an (explicit) local constancy result with respect to $a_p$ using Fontaine's theory of $(φ, Γ)$-modules and its crystalline refinement due to Berger via Wach modules and their continuity properties. The local constancy result with respect to $k$ (for $a_p\not=0$) will follow from a local study of Colmez's rigid analytic space parametrizin…
Irreducible crystalline representations of dimension 2 of Gal(Qpbar/Qp) depend up to twist on two parameters, the weight k and the trace of frobenius a_p. We show that the reduction modulo p of such a representation is a locally constant function of a_p (with an explicit radius) and a locally constant function of the weight k if a_p <> 0. We then give an algorithm for computing the reductions modulo p of these representations. The main ingredient is Fontaine's theory of (phi,Gamma)-modules as well as the theory of Wach modules.
We consider the family of irreducible crystalline representations of dimension $2$ of ${\rm Gal}(\overline{\bf Q}_p/{\bf Q}_p)$ given by the $V_{k,a_p}$ for a fixed weight integer $k\geq 2$. We study the locus of the parameter $a_p$ where these representations have a given reduction modulo $p$. We give qualitative results on this locus and show that for a fixed $p$ and $k$ it can be computed by determining the reduction modulo $p$ of $V_{k,a_p}$ for a finite number of values of the parameter $a_p$. We also generalize these results to other Galois types.
We provide a simple approach for the crystalline comparison of Ainf-cohomology, and reprove the comparison between crystalline and p-adic etale cohomology for formal schemes in the case of good reduction.
We prove some new cases of weight part of Serre's conjectures for mod $p$ Galois representations associated to automorphic representations on unitary groups $U(d)$. The approach is a generalization of the work of Gee-Liu-Savitt, namely, we study reductions of certain crystalline representations, as well as crystalline lifts of these reductions.
A conjecture of Breuil, Buzzard, and Emerton says that the slopes of certain reducible $p$-adic Galois representations must be integers. In previous work we showed this conjecture for representations that lie over certain non-subtle components of weight space. This article is a continuation of that work in which we also show the conjecture for the subtle components for slopes less than $\frac{p-1}{2}$.
We prove several results concerning the existence of potentially crystalline lifts with prescribed Hodge-Tate weights and inertial types of a given n-dimensional mod p representation of the absolute Galois group of K, where K/Q_p is a finite extension. Some of these results are proved by purely local methods, and are expected to be useful in the application of automorphy lifting theorems. The proofs of the other results are global, making use of automorphy lifting theorems.
In this paper, we study the Tamagawa numbers of a crystalline representation over a tower of cyclotomic extensions under certain technical conditions on the representation. In particular, we show that we may improve the asymptotic bounds given in the thesis of Arthur Laurent in certain cases.
Let $K$ be a field of characteristic zero complete for a discrete valuation, with perfect residue field of characteristic $p>0$, and let $K^+$ be the valuation ring of $K$. We relate the log-crystalline cohomology of the special fibre of certain affine $K^+$-schemes $X=\text{Spec}(R)$ with semi-stable reduction to the Galois cohomology of the fundamental group of the geometric generic fibre $π_1(X_{\bar{K}})$ with coefficients in a Fontaine ring constructed from $R$. This is based on Faltings' theory of almost étale extensions.
Mark Kisin proved that a certain restriction functor on crystalline p-adic representations is fully faithful. In this paper, we prove the torsion analogue of Kisin's theorem.
Let $p>2$ be a prime. Let $ρ$ be a crystalline representation of $G_{\mathbb{Q}_p}$ with distinct Hodge-Tate weights in $[0, p]$, such that its reduction $\overline ρ$ is upper triangular. Under certain conditions, we prove that $\overline ρ$ has an upper triangular crystalline lift $ρ'$ such that $\mathrm{HT}(ρ')=\mathrm{HT}(ρ)$. The method is based on the author's previous work, combined with an inspiration from the work of Breuil-Herzig.
Let K be a finite unramified extension of Q_p. We parametrize the (phi, Gamma)-modules corresponding to reducible two-dimensional mod p representations of G_K and characterize those which have reducible crystalline lifts with certain Hodge-Tate weights.
We prove a zig-zag conjecture describing the reductions of irreducible crystalline two-dimensional representations of $G_{\mathbb{Q}_p}$ of slope $\frac{3}{2}$ and exceptional weights. This along with previous works completes the description of the reduction for all slopes less than $2$. The proof involves computing the reductions of the Banach spaces attached by the $p$-adic LLC to these representations, followed by an application of the mod $p$ LLC to recover the reductions of these representations.
We describe the semi-simplification of the mod $p$ reduction of certain crystalline two dimensional local Galois representations of slopes in the interval $(1,2)$ and all weights. The proof uses the mod $p$ Local Langlands Correspondence for $GL_2(Q_p)$. We also give a complete description of the submodules generated by the second highest monomial in the mod $p$ symmetric power representations of $GL_2(F_p)$.
Let K be a p-adic local field with residue field k such that [k:k^p]=p^e<\infty and V be a p-adic representation of Gal(\bar{K}/K). Then, by using the theory of p-adic differential modules, we show that V is a potentially crystalline (resp. potentially semi-stable) representation of Gal(\bar{K}/K) if and only if V is a potentially crystalline (resp. potentially semi-stable) representation of Gal(\bar{K^{pf}}/K^{pf}) where K^{pf}/K is a certain p-adic local field whose residue field is the smallest perfect field k^{pf} containing k. As an application, we prove the p-adic monodromy theorem of Fo…
Let $p$ be a prime, let $K$ be a discretely valued extension of $\mathbb{Q}_p$, and let $A_{K}$ be an abelian $K$-variety with semistable reduction. Extending work by Kim and Marshall from the case where $p>2$ and $K/\mathbb{Q}_p$ is unramified, we prove an $l=p$ complement of a Galois cohomological formula of Grothendieck for the $l$-primary part of the Néron component group of $A_{K}$. Our proof involves constructing, for each $m\in \mathbb{Z}_{\geq 0}$, a finite flat $\mathscr{O}_K$-group scheme with generic fiber equal to the maximal 1-crystalline submodule of $A_{K}[p^{m}]$. As a corollar…
We determine reductions of 2-dimensional, irreducible, semi-stable, and non-crystalline representations of $\mathrm{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)$ with Hodge--Tate weights $0 < k-1$ and with $\mathcal L$-invariant whose $p$-adic norm is sufficiently large, depending on $k$. Our main result provides the first systematic examples of the reductions for $k \geq p$.
We construct explicitly some analytic families of etale (phi,Gamma)-modules, which give rise to analytic families of 2-dimensional crystalline representations. As an application of our constructions, we verify some conjectures of Breuil on the reduction modulo p of those representations, and extend some results (of Deligne, Edixhoven, Fontaine and Serre) on the representations arising from modular forms.
We show that the universal unitary completion of certain locally algebraic representation of $G:=\GL_2(\Qp)$ with $p>2$ is non-zero, topologically irreducible, admissible and corresponds to a 2-dimensional crystalline representation with non-semisimple Frobenius via the $p$-adic Langlands correspondence for $G$.
This paper studies crystalline representations of G_K with coefficients of any dimension, where K is the unramified extension of Q_p of degree a. We prove a theorem of Fontaine-Laffaille type when σ-invariant Hodge-Tate weight less than p-1, which establishes the bijection between Galois stable lattices in crystalline representations and strongly divisible ϕ-lattice. In generalizing Breuil's work, we classify all reducible and irreducible crystalline representations of G_K of dimensional 2, then describe their mod p reductions. We generalize some results (of Deligne, Fontaine-Serre, and Edixho…
Let $p>3$ be a prime number and let $G_{\mathbb{Q}_p}$ be the absolute Galois group of $\mathbb{Q}_p$. In this paper, we find Galois stable lattices in the irreducible $3$-dimensional semi-stable and non-crystalline representations of $G_{\mathbb{Q}_p}$ with Hodge--Tate weights $(0,1,2)$ by constructing their strongly divisible modules. We also compute the Breuil modules corresponding to the mod $p$ reductions of the strongly divisible modules, and determine which of the semi-stable representations has an absolutely irreducible mod $p$ reduction.
We construct the crystalline fundamental group of a semi-stable variety over a field of positive characteristic using the log De Rham-Witt complex and Navarro-Aznar's derived Thom-Whitney functor. This approach gives a relatively direct comparison theorem with the De Rham fundamental group in the liftable case. Also, we get a natural weight filtration which makes the coordinate ring of the crystalline fundamental group into a mixed isocrystal when the variety is defined over a finite field. Finally, the monodromy operator on the crystalline fundamental group is used to prove a crystalline anal…
We compute the reduction modulo $p$ of 2-dimensional crystalline representations whose Hodge-Tate weights are 0 and $k-1$ with $k \in \{p+2,...,2p-1\}$.
We compute the reductions of irreducible crystalline two-dimensional representations of $G_{\mathbf{Q}_p}$ of slope 1, for primes $p \geq 5$, and all weights. We describe the semisimplification of the reductions completely. In particular, we show that the reduction is often reducible. We also investigate whether the extension obtained is peu or très ramifiée, in the relevant reducible non-semisimple cases. The proof uses the compatibility between the $p$-adic and mod $p$ Local Langlands Correspondences, and involves a detailed study of the reductions of both the standard and non-standard latti…
Let $p$ be a prime, $K$ a finite extension of $\mathbb Q_p$, and let $G_K$ be the absolute Galois group of $K$. The category of étale $(φ, τ)$-modules is equivalent to the category of $p$-adic Galois representations of $G_K$. In this paper, we show that all étale $(φ, τ)$-modules are overconvergent; this answers a question of Caruso. Our result is an analogy of the classical overconvergence result of Cherbonnier and Colmez in the setting of étale $(φ, Γ)$-modules. However, our method is completely different from theirs. Indeed, we first show that all $p$-power-torsion representations admit loo…
We propose the notion of the {\em crystalline sub-representation functor} defined on $p$-adic representations of the Galois groups of finite extensions of $\Qp$, with certain restrictions in the case of integral representations. By studying its right-derived functors, we find a natural extension of a formula of Grothendieck expressing the group of connected components of a Neron model of an abelian variety in terms of Galois cohomology.
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…
Let $p>2$ be a prime. Let $K$ be a tamely ramified finite extension over $\mathbb Q_p$ with ramification index $e$, and let $G_K$ be the Galois group. We study Kisin modules attached to crystalline representations of $G_K$ whose labeled Hodge-Tate weights are relatively small (a sort of "$er\le p$" condition where $r$ is the maximal Hodge-Tate weight). In particular, we show that these Kisin modules admit "adapted bases". We then apply these results in the special case $e=2$ to study reductions and liftings of certain crystalline representations. As a consequence, we establish some new cases o…
In this article we study local constancy of the mod $p$ reduction of certain $2$-dimensional crystalline representations of $\mathrm{Gal}\left(\bar{\mathbb{Q}}_p/\mathbb{Q}_p\right)$ using the mod $p$ local Langlands correspondence. We prove local constancy in the weight space by giving an explicit lower bound on the local constancy radius centered around weights going up to $(p-1)^{2} +3$ and the slope fixed in $(0, \ p-1)$ satisfying certain constraints. We establish the lower bound by determining explicitly the mod $p$ reductions at nearby weights and applying a local constancy result of Be…
Let $\mathbf{B}(V)$ be the admissible unitary $GL_2(\mathbb{Q}_p)$-representation associated to two dimensional crystalline Galois representation $V$ by $p$-adic Langlands constructed by Breuil. Berger and Breuil conjectured an explicit description of the locally analytic vectors $\mathbf{B}(V)_{\mathrm{la}}$ of $\mathbf{B}(V)$ which is now proved by Liu. Emerton recently studied $p$-adic representations from the viewpoint of rigid analytic geometry. In this article, we consider certain rigid analytic subgroups of $GL(2)$ and give an explicit description of the rigid analytic vectors in $\math…
We show the existence of integral models for cuspidal representations of GL(2,q), whose reduction modulo p can be identified with the cokernel of a differential operator on F_{q}[X,Y] defined by J-P. Serre. These integral models come from the crystalline cohomology of the projective curve XY^{q}-X^{q}Y-Z^{q+1}=0. As an application, we can extend a construction of C. Khare and B. Edixhoven (2003) giving a cohomological analogue of the Hasse invariant operator acting on spaces of modp modular forms for GL(2).
Let k be an algebraically closed field of characteristic p>0. Let W(k) be the ring of Witt vectors with coefficients in k. We prove a motivic conjecture of Milne that relates, in the case of abelian schemes, the étale cohomology with $\dbZ_p$ coefficients to the crystalline cohomology with integral coefficients, in the more general context of p-divisible groups endowed with {\it arbitrary} families of crystalline tensors over a finite, discrete valuation ring extension of W(k). This extends a result of Faltings in [Fa2]. As a main new tool we construct global deformations of p-divisible groups…
