scholarly journals Deformation Theory of the Trivial mod p Galois Representation for GLn

2020 ◽  
Author(s):  
Ashwin Iyengar
Keyword(s):  
Deformation Theory ◽  
Finite Extension ◽  
Galois Representation ◽  
Deformation Space ◽  
Absolute Galois Group ◽  
Trivial Representation ◽  
Characteristic P ◽  
Generic Fiber ◽  
Mild Conditions ◽  
Irreducible Components

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

scholarly journals Density of potentially crystalline representations of fixed weight

2016 ◽  
Vol 152 (8) ◽  
pp. 1609-1647 ◽  
Author(s):  
Eugen Hellmann ◽  
Benjamin Schraen
Keyword(s):  
Galois Representations ◽  
Finite Extension ◽  
Abelian Extension ◽  
Unitary Groups ◽  
Absolute Galois Group ◽  
Generic Fiber ◽  
Density Result ◽  
Crystalline Representations ◽  
Fixed Weight ◽  
Similar Density

Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and let $\bar{\unicode[STIX]{x1D70C}}$ be a continuous, absolutely irreducible representation of its absolute Galois group with values in a finite field of characteristic $p$. We prove that the Galois representations that become crystalline of a fixed regular weight after an abelian extension are Zariski-dense in the generic fiber of the universal deformation ring of $\bar{\unicode[STIX]{x1D70C}}$. In fact we deduce this from a similar density result for the space of trianguline representations. This uses an embedding of eigenvarieties for unitary groups into the spaces of trianguline representations as well as the corresponding density claim for eigenvarieties as a global input.

Locally indecomposable Galois representations with full residual image

2017 ◽  
Vol 13 (05) ◽  
pp. 1191-1211
Author(s):  
Devika Sharma
Keyword(s):  
Modular Forms ◽  
Deformation Theory ◽  
Galois Representations ◽  
Galois Representation ◽  
Mild Conditions ◽  
Residual Image ◽  
Hida Families

We consider certain [Formula: see text]-ordinary non-CM Hida families with full residual Galois representation and give mild conditions under which every arithmetic point in these families is locally indecomposable when [Formula: see text]. The proof uses methods from deformation theory and mostly works for any odd prime [Formula: see text], but ultimately relies on the existence of a weight [Formula: see text] form in an auxiliary family which is available only for [Formula: see text]. We end by giving several non-trivial examples of [Formula: see text]-ordinary non-CM locally indecomposable modular forms of small level with full residual Galois representation.

scholarly journals FAMILIES OF AFFINE RULED SURFACES: EXISTENCE OF CYLINDERS

2016 ◽  
Vol 223 (1) ◽  
pp. 1-20 ◽  
Author(s):  
ADRIEN DUBOULOZ ◽  
TAKASHI KISHIMOTO
Keyword(s):  
Minimal Model ◽  
Finite Extension ◽  
Smooth Curve ◽  
Ruled Surfaces ◽  
Model Program ◽  
Minimal Model Program ◽  
Generic Fiber ◽  
Projective Model ◽  
The Family

We show that the generic fiber of a family $f:X\rightarrow S$ of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a finite extension of the base $S$. In the particular case where the general fibers of the family are irrational surfaces, we establish that up to shrinking $S$, such a family actually factors through an $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ over a certain $S$-scheme $Y\rightarrow S$ induced by the MRC-fibration of a relative smooth projective model of $X$ over $S$. For affine threefolds $X$ equipped with a fibration $f:X\rightarrow B$ by irrational $\mathbb{A}^{1}$-ruled surfaces over a smooth curve $B$, the induced $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ can also be recovered from a relative minimal model program applied to a smooth projective model of $X$ over $B$.

Independence of ℓ \ell -adic representations of geometric Galois groups

2018 ◽  
Vol 2018 (736) ◽  
pp. 69-93 ◽  
Author(s):  
Gebhard Böckle ◽  
Wojciech Gajda ◽  
Sebastian Petersen
Keyword(s):  
Galois Group ◽  
Finite Type ◽  
Algebraic Closure ◽  
Finite Extension ◽  
Field Extension ◽  
Galois Groups ◽  
Closed Field ◽  
Absolute Galois Group ◽  
Arbitrary Characteristic ◽  
Cohomology Modules

AbstractLetkbe an algebraically closed field of arbitrary characteristic, let{K/k}be a finitely generated field extension and letXbe a separated scheme of finite type overK. For each prime{\ell}, the absolute Galois group ofKacts on the{\ell}-adic étale cohomology modules ofX. We prove that this family of representations varying over{\ell}is almost independent in the sense of Serre, i.e., that the fixed fields inside an algebraic closure ofKof the kernels of the representations for all{\ell}become linearly disjoint over a finite extension ofK. In doing this, we also prove a number of interesting facts on the images and on the ramification of this family of representations.

scholarly journals Structure and classification of Hom-associative algebras

2020 ◽  
Vol 24 (1) ◽  
pp. 79-102
Author(s):  
Abdenacer Makhlouf ◽  
Ahmed Zahari
Keyword(s):  
Deformation Theory ◽  
Matrix Algebra ◽  
Algebraic Varieties ◽  
Associative Algebras ◽  
Irreducible Components

The purpose of this paper is to study the structure and the algebraic varieties of Hom-associative algebras. We characterize multiplicative simple Hom-associative algebras and give some examples deforming the 2 × 2-matrix algebra to simple Hom-associative algebras. We provide a classification of n-dimensional Hom-associative algebras for n ≤ 3. Then we study irreducible components using deformation theory.

Characteristic p Galois Representations That Arise from Drinfeld Modules

2000 ◽  
Vol 43 (3) ◽  
pp. 282-293 ◽  
Author(s):  
Nigel Boston ◽  
David T. Ose
Keyword(s):  
Finite Field ◽  
Galois Group ◽  
Galois Representations ◽  
Drinfeld Module ◽  
Drinfeld Modules ◽  
Absolute Galois Group ◽  
Characteristic P ◽  
Finite Characteristic ◽  
The Absolute

AbstractWe examine which representations of the absolute Galois group of a field of finite characteristic with image over a finite field of the same characteristic may be constructed by the Galois group’s action on the division points of an appropriate Drinfeld module.

scholarly journals VARIÉTÉS DE KISIN STRATIFIÉES ET DÉFORMATIONS POTENTIELLEMENT BARSOTTI-TATE

2016 ◽  
Vol 17 (5) ◽  
pp. 1019-1064 ◽  
Author(s):  
Xavier Caruso ◽  
Agnès David ◽  
Ariane Mézard
Keyword(s):  
Galois Group ◽  
Finite Extension ◽  
Finite Union ◽  
Two Dimensional ◽  
Explicit Computation ◽  
Absolute Galois Group ◽  
Dimensional Representation ◽  
The Absolute

Let $F$ be a unramified finite extension of $\mathbb{Q}_{p}$ and $\overline{\unicode[STIX]{x1D70C}}$ be an irreducible mod $p$ two-dimensional representation of the absolute Galois group of $F$. The aim of this article is the explicit computation of the Kisin variety parameterizing the Breuil–Kisin modules associated to certain families of potentially Barsotti–Tate deformations of $\overline{\unicode[STIX]{x1D70C}}$. We prove that this variety is a finite union of products of $\mathbb{P}^{1}$. Moreover, it appears as an explicit closed connected subvariety of $(\mathbb{P}^{1})^{[F:\mathbb{Q}_{p}]}$. We define a stratification of the Kisin variety by locally closed subschemes and explain how the Kisin variety equipped with its stratification may help in determining the ring of Barsotti–Tate deformations of $\overline{\unicode[STIX]{x1D70C}}$.

A local-to-global principle for deformations of Galois representations

1999 ◽  
Vol 1999 (509) ◽  
pp. 199-236 ◽  
Author(s):  
Gebhard Böckle
Keyword(s):  
Finite Field ◽  
Galois Representations ◽  
Galois Representation ◽  
Local Deformation ◽  
Galois Groups ◽  
Equivalence Classes ◽  
Deformation Space ◽  
Finite Set ◽  
Global Deformation ◽  
Global Principle

Abstract Given an absolutely irreducible Galois representation : GE → GLN (k), E a number field, k a finite field of characteristic l > 2, and a finite set of places Q of E containing all places above l and ∞ and all where ∞ ramifies, there have been defined many functors representing strict equivalence classes of deformations of such a representation, e.g. by Mazur or Wiles in [15] or [26], with various conditions on the behaviour of the deformations at the places in Q and with the condition that the deformations are unramified outside Q. Those functors are known to be representable. For as above, our goal is to present a rather general class of global deformation functors that satisfy local deformation conditions and to investigate for those, under what conditions the global deformation functor is determined by the local deformation functors. We will give precise conditions under which the local functors for all places in Q are sufficient to describe the global functor, first in a coarse form, then in a refined form using auxiliary primes as done by Taylor and Wiles in [24]. This has several consequences. The strongest is that one can derive ring theoretic results for the universal deformation space by Mazur if one uses results of Diamond and Wiles, cf. [11] and [26], and if one has a good understanding of all local situations. Furthermore it is easier to understand what happens under increasing the ramification as done by Boston and Ramakrishna in [6] and [20], [21]. Finally we shall reinterpret the results in the case of a tame representation by directly considering presentations of certain pro-l Galois groups and revisiting the prime-to-adjoint principle of Boston, cf. [5].

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