On reductions of families of crystalline Galois representations Part II

Galois families of modular forms and application to weight one

We introduce Galois families of modular forms. They are a new kind of family coming from Galois representations of the absolute Galois groups of rational function fields over the rational field. We exhibit some examples and provide an infinite Galois family of non-liftable weight one Katz modular eigenforms over an algebraic closure of F_p for p in {3,5,7,11}.

The Grunwald problem and specialization of families of regular Galois extensions

We investigate specializations of infinite families of regular Galois extensions over number fields. The problem to what extent the local behaviour of specializations of one single regular Galois extension can be prescribed has been investigated by Dèbes and Ghazi in the unramified case, and by Legrand, Neftin and the author in general. Here, we generalize these results and give a partial solution to Grunwald problems using Galois extensions arising as specializations of a family of regular Galois extensions.

A control theorem for the images of Galois actions on certain infinite families of modular forms

In this letter we show that for certain infinite families of modular forms of growing level it is possible to have a control result for the exceptional primes of the attached Galois representations. As an application, a uniform version of a result of Wiese on Galois realizations of linear groups over finite fields of arbitrarily large exponent is proved.

Lyndon-Demushkin 方法与 $G_2$ 值 Galois 表示的晶体提升

Lyndon-Demushkin method and crystalline lifts of $G_2$-valued Galois representations

本文证明对于所有局部域（p>3），所有 G_2 值 Galois 表示均存在晶体提升，主要利用组合群论新技术分析非交换扩张障碍，并借助 Emerton-Gee 栈构造交换扩张。

On the generalized Brauer-Siegel theorem for asymptotically exact families with solvable Galois closure

In 2002, M. A. Tsfasman and S. G. Vlăduţ formulated the generalized Brauer-Siegel conjecture for asymptotically exact families of number fields. In this article, we establish this conjecture for asymptotically good towers and asymptotically bad families of number fields with solvable normal closure.

An approach for computing families of multi-branch-point covers and applications for symplectic Galois groups

We propose an approach for the computation of multi-parameter families of Galois extensions with prescribed ramification type. More precisely, we combine existing deformation and interpolation techniques with recently developed strong tools for the computation of $3$-point covers. To demonstrate the applicability of our method in relatively large degrees, we compute several families of polynomials with symplectic Galois groups, in particular obtaining the first totally real polynomials with Galois group PSp(6,2).

Geometric families of 4-dimensional Galois representations with generically large images

We study compatible families of four-dimensional Galois representations constructed in the étale cohomology of a smooth projective variety. We prove a theorem asserting that the images will be generically large if certain conditions are satisfied. We only consider representations with coefficients in an imaginary quadratic field. We apply our result to an example constructed by Jasper Scholten, obtaining a family of linear groups and one of unitary groups as Galois groups over $\mathbb{Q}$.

Purity for families of Galois representations

We formulate a notion of purity for $p$-adic big Galois representations and pseudorepresentations of Weil groups of $\ell$-adic number fields for $\ell\neq p$. This is obtained by showing that all powers of the monodromy of any big Galois representation stay "as large as possible" under pure specializations. The role of purity for families in the study of the variation of local Euler factors, local automorphic types along irreducible components, the intersection points of irreducible components of $p$-adic families of automorphic Galois representations is illustrated using the examples of Hida…

棱晶 $F$-晶体与晶体 Galois 表示

Prismatic $F$-crystals and crystalline Galois representations

本文证明了在混合特征完备离散赋值域上，棱晶 $F$-晶体范畴与晶体 $G_K$-表示中的格范畴是等价的。

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 $(φ,τ)$-module…

Deformations of Reducible Galois Representations to Hida-Families

The global deformation theory of residually reducible Galois representations with fixed auxiliary conditions is studied. We show that $\barρ:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}_2(\bar{\mathbb{F}}_p)$ lifts to a Hida line for which the weights range over a congruence class modulo-$p^2$. The advantage of the purely Galois theoretic approach is that it allows us to construct $p$-adic families of Galois representations lifting the actual representation $\barρ$, and not just the semisimplification.

具有固定行列式的半单 $G$ 值伽罗瓦表示的结晶提升

Crystalline lifts of semisimple $G$-valued Galois representations with fixed determinant

本文证明了对于有限扩张和分裂约化群，拟半单 mod $p$ $G$ 值伽罗瓦表示在固定阿贝尔化结晶提升下存在具有正则 Hodge-Tate 权重的结晶提升，并将该结果推广至拟分裂驯顺群和半单 $L$-参数情形。

Existence of families of Galois representations and new cases of the Fontaine-Mazur conjecture

In a previous article, we have proved a result asserting the existence of a compatible family of Galois representations containing a given crystalline irreducible odd two-dimensional representation. We apply this result to establish new cases of the Fontaine-Mazur conjecture, namely, an irreducible Barsotti-Tate $λ$-adic 2-dimensional Galois representation unramified at 3 and such that the traces $a_p$ of the images of Frobenii verify $\Q(\{a_p^2 \}) = \Q $ always comes from an abelian variety. We also show the non-existence of irreducible Barsotti-Tate 2-dimensional Galois representations of …

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…

小抛物特征形式上Siegel亏格为2的模形式的辛Galois表示族

Families of symplectic Galois representations over small parabolic eigenvarieties for Siegel cuspforms of genus $2$

本文构造了小抛物特征形式并研究其附带的辛Galois表示族，引入了带有G结构的(φ, Γ)-模和精炼的辛Galois表示族理论，并证明了在温和假设下的微分R=T定理，探讨了小抛物特征形式在Saito-Kurokawa提升与Bloch-Kato Selmer群的关系。

Uniform behavior of families of Galois representations on Siegel modular forms and the Endoscopy Conjecture

We prove the following uniformity principle: if one of the Galois representations in the family attached to a genus two Siegel cusp form of weight $k>3$, "semistable" and with multiplicity one, is reducible (for an odd prime $p$),then all the representations in the family are reducible. This, combined with Serre's conjecture (which is now a theorem) gives a proof of the Endoscopy Conjecture.

On the classification of geometric families of 4-dimensional Galois representations

We give a classification theorem for certain four-dimensional families of geometric $λ$-adic Galois representations attached to a pure motive. More precisely, we consider families attached to the cohomology of a smooth projective variety defined over $\Q$ with coefficients in a quadratic imaginary field, non-selfdual and with four different Hodge-Tate weights. We prove that the image is as large as possible for almost every $λ$ provided that the family is irreducible and not induced from a family of smaller dimension. If we restrict to semistable families an even simpler classification is give…

Galois groups arising from families with big orthogonal monodromy

We study the Galois groups of polynomials arising from a compatible family of representations with big orthogonal monodromy. We show that the Galois groups are usually as large as possible given the constraints imposed on them by a functional equation and discriminant considerations. As an application, we consider the Frobenius polynomials arising from the middle étale cohomology of hypersurfaces in $\mathbb{P}_{\mathbb{F}_q}^{2n+1}$ of degree at least $3$. We also consider the $L$-functions of quadratic twists of fixed degree of an elliptic curve over a function field $\mathbb{F}_q(t)$. To de…

模 Galois 表示族中的 $λ$-不变稳定性

$λ$-invariant stability in families of modular Galois representations

本文从定量角度研究权为 2 的模形式族中 Iwasawa $λ$-不变量的变化，提供了 $λ$-不变量增长或保持稳定的频率下界。

通过 Selmer 群的分布研究扭族中的 Galois 模结构与 Hasse 原理

Galois module structures and the Hasse principle in twist families via the distribution of Selmer groups

本文利用 Abel 簇稀疏二次扭族中 2-Selmer 秩的分布结果，统一研究 Mordell–Weil 群 Galois 模结构及 Hasse 原理失效等问题。

p-adic families of Galois representations and higher rank Selmer groups

This is the final version of a book about p-adic families of Galois representations, Selmer groups, eigenvarieties and Arthur's conjectures

Big image of Galois representations associated with finite slope $p$-adic families of modular forms

We consider the Galois representation associated with a finite slope $p$-adic family of modular forms. We prove that the Lie algebra of its image contains a congruence Lie subalgebra of a non-trivial level. We describe the largest such level in terms of the congruences of the family with $p$-adic CM forms.

Families of p-adic Galois representations and (phi,Gamma)-modules

We investigate the relation between p-adic Galois representations and overconvergent (phi,Gamma)-modules in families. Especially we construct a natural open subspace of a family of (phi,Gamma)-modules, over which it is induced by a family of Galois-representations.

Two families of pro-p groups that are not absolute Galois groups

Let $p$ be a prime. We produce two new families of pro-$p$ groups which are not realizable as absolute Galois groups of fields. To prove this we use the 1-smoothness property of absolute Galois pro-$p$ groups. Moreover, we show in these families one has one-relator pro-$p$ groups which may not be ruled out as absolute Galois groups employing the quadraticity of Galois cohomology (a consequence of Rost-Voevodsky Theorem), or the vanishing of Massey products in Galois cohomology.

棱镜 $F$-晶体与 $E$-晶体伽罗瓦表示

Prismatic $F$-crystals and $E$-crystalline Galois representations

本文证明棱镜 $F$-晶体范畴与 $E$-晶体伽罗瓦表示的 $\mathcal{O}_E$-格范畴等价，推广了 $E=\mathbf{Q}_p$ 情形，并通过改编 Du--Liu 引理简化了下降步骤。

Limit of torsion semi-stable Galois representations with unbounded weights

Let $K$ be a complete discrete valuation field of characteristic $(0, p)$ with perfect residue field, and let $T$ be an integral $\mathbb{Z}_p$-representation of $\mathrm{Gal}(\overline{K}/K)$. A theorem of T. Liu says that if $T/p^n T$ is torsion semi-stable (resp. crystalline) of uniformly bounded Hodge-Tate weights for all $n \geq 1$, then $T$ is also semi-stable (resp. crystalline). In this note, we show that we can relax the condition of "uniformly bounded Hodge-Tate weights" to an unbounded (log-)growth condition.

A Note on Potential Diagonalizability of Crystalline Representations

Let K_0 be a finite unramified extension of Q_p. We show that all crystalline representations of G_{K_0} (the absolute Galois group of K_0) with Hodge-Tate weights in {0, ..., p-1} are potentially diagonalizable.

通过μ_p-等变性对晶体伽罗瓦表示进行模p约化

Reduction modulo p of crystalline Galois representations via μ_p-equivariance

研究\(\mathbb{Q}_p\)绝对伽罗瓦群晶体表示模p约化的惯性权重，通过棱柱技术使布雷伊尔 - 基辛模模p约化获\(\mu_p\)-等变结构，结合仿射格拉斯曼流形结果，明确塞尔权重猜想并证其消除方向。

斜率(2,3)的模Galois表示的约化

Reductions of modular Galois representations of Slope (2,3)

本文基于Bhattacharya-Ghate的工作，计算了p-adic数绝对Galois群的二维结晶表示在斜率(2,3)及任意权重下模p约化的半简化。

Families of abelian varieties and large Galois images

Associated to an abelian variety $A$ of dimension $g$ over a number field $K$ is a Galois representation $ρ_A\colon Gal(\bar{K}/K)\to GL_{2g}(\hat{\mathbb{Z}})$. The representation $ρ_A$ encodes the Galois action on the torsion points of $A$ and its image is an interesting invariant of $A$ that contains a lot of arithmetic information. We consider abelian varieties over $K$ parametrized by the $K$-points of a nonempty open subvariety $U\subseteq \mathbb{P}^n_K$. We show that away from a set of density $0$, the image of $ρ_A$ will be very large; more precisely, it will have uniformly bounded in…

On lifting and modularity of reducible residual Galois representations over imaginary quadratic fields

In this paper we study deformations of mod $p$ Galois representations $τ$ (over an imaginary quadratic field $F$) of dimension $2$ whose semi-simplification is the direct sum of two characters $τ_1$ and $τ_2$. As opposed to our previous work we do not impose any restrictions on the dimension of the crystalline Selmer group $H^1_Σ(F, {\rm Hom}(τ_2, τ_1)) \subset {\rm Ext}^1(τ_2, τ_1)$. We establish that there exists a basis $\mathcal{B}$ of $H^1_Σ(F, {\rm Hom}(τ_2, τ_1))$ arising from automorphic representations over $F$ (Theorem 8.1). Assuming among other things that the elements of $\mathcal{…

Selmer秩在偶Galois表示族中的密度，Wiles公式与全球互反律

Density of Selmer ranks in families of even Galois representations, Wiles' formula, and global reciprocity

本文通过Galois上同调方法研究偶Galois表示族中Selmer秩的分布，利用Wiles公式和全球互反律证明了Selmer秩增加的素数密度为1/192。

Galois level and congruence ideal for $p$-adic families of finite slope Siegel modular forms

We consider $p$-adic families of Siegel eigenforms of genus $2$ and finite slope, defined as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the image of the Galois representation associated with a family is big, in the sense that a Lie algebra attached to it contains a congruence subalgebra of non-zero level. We call Galois level of the family the largest such level. We show that it is trivial when the residual representation has full image. When the residual representation is a symmetric cube, the zer…

A note on the crystalline subrepresentation functor

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.

Infinitely ramified Galois representations

In this paper we show how to construct, for most p >= 5, two types of surjective representations ρ:G_Q=Gal(\bar{Q}/Q) -> GL_2(Z_p) that are ramified at an infinite number of primes. The image of inertia at almost all of these primes will be torsion-free. The first construction is unconditional. The catch is that we cannot say whether ρ|_{G_p=Gal(\bar{Q_p}/Q_p) is crystalline or even potentially semistable. The second construction assumes the Generalized Riemann Hypothesis (GRH). With this assumption we can further arrange that ρ|_{G_p} is crystalline at p. We remark that infinitely ramified *r…

Crystalline liftings and weight part of Serre's conjecture

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.

非分歧情形下某些晶体表示的限制

Reductions of Some Crystalline Representations in the Unramified Setting

本文在非分歧情形下通过显式描述强可除格和构造典范 Kisin 模，确定了二维晶体表示的半单约化，并在参数 p-进赋值足够大时给出任意 Hodge-Tate 权下的约化结果及 Berger-Li-Zhu 型界。

crystalline points的Zariski密度

Zariski density of crystalline points

本文证明了crystalline points在p-adic域绝对Galois群表示的形变空间中具有Zariski稠密性，该纯局部证明适用于所有p-adic域和剩余Galois表示。

Expander graphs, gonality and variation of Galois representations

We show that families of coverings of an algebraic curve where the associated Cayley-Schreier graphs form an expander family exhibit strong forms of geometric (genus and gonality) growth. Combining this general result with finiteness statements for rational points under such conditions, we derive results concerning the variation of Galois representations in one-parameter families of abelian varieties.

Reductions of Galois representations for slopes in $(1,2)$

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 $GL_2(F_p)$.

Loose crystalline lifts and overconvergence of étale $(φ, τ)$-modules

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…

A class of p-adic Galois representations arising from abelian varierties over Q_p

Let V be a p-adic representation of the absolute Galois group G of Q_p that becomes crystalline over a finite tame extension, and assume p odd. We provide necessary and sufficient conditions for V to be isomorphic to the Tate module V_p(A) of an abelian variety A defined over Q_p. These conditions are stated on the filtered (ϕ,G)-module attached to V.

Semistable periods of finite slope families

We introduce the notion of finite slope families to encode the local properties of the p-adic families of Galois representations appearing in the work of Harris, Lan, Taylor and Thorne on the construction of Galois representations for (non-self dual) regular algebraic cuspidal automorphic representations of GL(n) over CM fields. Our main result is to prove the analytic continuation of semi-stable (and crystalline) periods for such families.

Several variable p-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations

Let F be a totally real field and G=GSp(4)_{/F}. In this paper, we show under a weak assumption that, given a Hecke eigensystem lambda which is (p,P)-ordinary for a fixed parabolic P in G, there exists a several variable p-adic family underline{lambda} of Hecke eigensystems (all of them (p,P)-nearly ordinary) which contains lambda. The assumption is that lambda is cohomological for a regular coefficient system. If F=Q, the number of variables is three. Moreover, in this case, we construct the three variable p-adic family rho_{underline{lambda}} of Galois representations associated to underline…

同余伽罗瓦表示族中的Selmer稳定性

Selmer stability in families of congruent Galois representations

本文研究同余模形式族中Selmer群的变化，探讨在固定素数p≥5下，通过Greenberg局部条件定义的Selmer群稳定性。证明在某些条件下，满足p-秩稳定性的模形式数量至少增长为X(logX)^{α-1}。

显式计算曲线族中出现的Galois表示

Explicit computation of Galois representations occurring in families of curves

本文将计算曲线Jacobian除法多项式的方法扩展至Q(t)上的曲线，以计算曲面étale上同调中的Galois表示，获得了P^1_Q上Galois表示的显式族并研究了其在坏约化处的退化。

伽罗瓦扩张族中Frobenius元素的分布

Distribution of Frobenius elements in families of Galois extensions

该研究通过伽罗瓦群表示与分歧理论不变量刻画伽罗瓦扩张族中Frobenius元素的精细分布，改进了GRH下Chebotarev密度定理误差项的波动上界，推广了相关经典结果并解决了公开问题。

An algorithm for computing the reduction of $2$-dimensional crystalline representations of $\text{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$

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.

Mod pq Galois representations and Serre's conjecture

Motives and automorphic forms of arithmetic type give rise to Galois representations that occur in {\it compatible families}. These compatible families are of p-adic representations with p varying. By reducing such a family mod p one obtains compatible families of mod p representations. While the representations that occur in such a p-adic or mod p family are strongly correlated, in a sense each member of the family reveals a new face of the motive. In recent celebrated work of Wiles playing off a pair of Galois representations in different characteristics has been crucial. In this paper we in…

On the reductions of certain two-dimensional crystalline representations

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 $\frac{p-1}{2}$.

二维中p有界晶态轨迹之间的包含关系

Inclusions between p-bounded crystalline loci in dimension two

该研究分析二维p有界晶态轨迹Z(r)的包含关系，除退化情形外通过三种Hodge型运算分类，除一例外可通过闭点包含检测，研究中大量使用GPT-5.5 Pro。

On some partially de Rham Galois representations

We study some partially de Rham representations of $\mathrm{Gal}(\bar{L}/L)$ for a finite unramified extension $L$ of $\mathbb{Q}_p$. We study some related subspaces of Galois cohomology and of cohomology of $B$-pairs. As an application, we associate a locally $\mathbb{Q}_p$-analytic representation of $\mathrm{GL}_2(L)$ to a $2$-dimensional trianguline representation of $\mathrm{Gal}(\bar{L}/L)$, by generalizing Breuil's theory in the crystalline case.

Reduction of certain crystalline representations and local constancy in the weight space

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.

Deformation theory of the trivial mod $p$ Galois representation for $\mathrm{GL}_n$

We study the rigid generic fiber $\mathcal{X}^\square_{\overlineρ}$ of the framed deformation space of the trivial representation $\overlineρ: G_K \to \text{GL}_n(k)$ where $k$ is a finite field of characteristic $p>0$ and $G_K$ is the absolute Galois group of a finite extension $K/\mathbf{Q}_p$. Under some mild conditions on $K$ we prove that $\mathcal{X}^\square_{\overlineρ}$ is normal. When $p > n$ we describe its irreducible components, and show Zariski density of its crystalline points.