Relaxation of strict parity for reducible Galois representations attached to the homology of GL(3, ℤ)

2016 ◽  
Vol 12 (02) ◽  
pp. 361-381 ◽  
Author(s):  
Avner Ash ◽  
Darrin Doud
Keyword(s):  
Irreducible Representation ◽  
Finite Field ◽  
Galois Group ◽  
Galois Representations ◽  
Algebraic Closure ◽  
Continuous Homomorphism ◽  
Two Dimensional ◽  
Absolute Galois Group ◽  
Direct Sum ◽  
Parity Condition

In this paper, we prove the following theorem: Let [Formula: see text] be an algebraic closure of a finite field of characteristic [Formula: see text]. Let [Formula: see text] be a continuous homomorphism from the absolute Galois group of [Formula: see text] to [Formula: see text]) which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. We assume that the image of [Formula: see text] is contained in the intersection of the stabilizers of the line spanned by [Formula: see text] and the plane spanned by [Formula: see text], where [Formula: see text] denotes the standard basis. Such [Formula: see text] will not satisfy a certain strict parity condition. Under the conditions that the Serre conductor of [Formula: see text] is squarefree, that the predicted weight [Formula: see text] lies in the lowest alcove, and that [Formula: see text], we prove that [Formula: see text] is attached to a Hecke eigenclass in [Formula: see text], where [Formula: see text] is a subgroup of finite index in [Formula: see text] and [Formula: see text] is an [Formula: see text]-module. The particular [Formula: see text] and [Formula: see text] are as predicted by the main conjecture of the 2002 paper of the authors and David Pollack, minus the requirement for strict parity.

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 Independence of -adic Galois representations over function fields

2013 ◽  
Vol 149 (7) ◽  
pp. 1091-1107 ◽  
Author(s):  
Wojciech Gajda ◽  
Sebastian Petersen
Keyword(s):  
Galois Group ◽  
Finite Type ◽  
Galois Representations ◽  
Function Fields ◽  
Algebraic Closure ◽  
Finite Extension ◽  
Prime Numbers ◽  
Positive Answer ◽  
Absolute Galois Group ◽  
The Family

AbstractLet$K$be a finitely generated extension of$\mathbb {Q}$. We consider the family of$\ell $-adic representations ($\ell $varies through the set of all prime numbers) of the absolute Galois group of$K$, attached to$\ell $-adic cohomology of a separated scheme of finite type over$K$. We prove that the fields cut out from the algebraic closure of$K$by the kernels of the representations of the family are linearly disjoint over a finite extension of K. This gives a positive answer to a question of Serre.

scholarly journals Minimally ramified deformations when

2018 ◽  
Vol 155 (1) ◽  
pp. 1-37 ◽  
Author(s):  
Jeremy Booher
Keyword(s):  
Finite Field ◽  
Galois Group ◽  
Galois Representations ◽  
Deformation Condition ◽  
Nilpotent Orbits ◽  
Absolute Galois Group ◽  
Unipotent Element ◽  
Change Type ◽  
Nilpotent Elements ◽  
The Absolute

Let $p$ and $\ell$ be distinct primes, and let $\overline{\unicode[STIX]{x1D70C}}$ be an orthogonal or symplectic representation of the absolute Galois group of an $\ell$-adic field over a finite field of characteristic $p$. We define and study a liftable deformation condition of lifts of $\overline{\unicode[STIX]{x1D70C}}$ ‘ramified no worse than $\overline{\unicode[STIX]{x1D70C}}$’, generalizing the minimally ramified deformation condition for $\operatorname{GL}_{n}$ studied in Clozel et al. [Automorphy for some$l$-adic lifts of automorphic mod$l$Galois representations, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181; MR 2470687 (2010j:11082)]. The key insight is to restrict to deformations where an associated unipotent element does not change type when deforming. This requires an understanding of nilpotent orbits and centralizers of nilpotent elements in the relative situation, not just over fields.

Nonexistence of certain Galois representations for quadratic fields

2018 ◽  
Vol 14 (05) ◽  
pp. 1505-1524
Author(s):  
Adam Gamzon ◽  
Lance Edward Miller
Keyword(s):  
Galois Group ◽  
Galois Representations ◽  
Number Fields ◽  
Quadratic Fields ◽  
Two Dimensional ◽  
Absolute Galois Group ◽  
Quadratic Number ◽  
Quadratic Number Fields

We establish new cases of quadratic number fields [Formula: see text] unramified away from a prime [Formula: see text] and [Formula: see text] whose absolute Galois group has no irreducible two-dimensional continuous Galois representations in [Formula: see text]. Our work builds on methods of Moon–Taguchi and Şengün and the usual analytic techniques of Odlyzko and Poitou where we note one of the new conditional cases arises via a correction of Poitou’s estimate. The results here seem optimal in that it seems these methods alone will yield no further cases either due to prohibitive computational issues or a failure of the analytic obstructions.

TORSION OF ABELIAN VARIETIES OVER LARGE ALGEBRAIC EXTENSIONS OF

2017 ◽  
Vol 234 ◽  
pp. 46-86
Author(s):  
MOSHE JARDEN ◽  
SEBASTIAN PETERSEN
Keyword(s):  
Galois Group ◽  
Abelian Variety ◽  
Haar Measure ◽  
Rational Point ◽  
Algebraic Closure ◽  
Prime Numbers ◽  
Absolute Galois Group ◽  
Finitely Generated ◽  
Fixed Field ◽  
Almost All

Let$K$be a finitely generated extension of$\mathbb{Q}$, and let$A$be a nonzero abelian variety over$K$. Let$\tilde{K}$be the algebraic closure of$K$, and let$\text{Gal}(K)=\text{Gal}(\tilde{K}/K)$be the absolute Galois group of$K$equipped with its Haar measure. For each$\unicode[STIX]{x1D70E}\in \text{Gal}(K)$, let$\tilde{K}(\unicode[STIX]{x1D70E})$be the fixed field of$\unicode[STIX]{x1D70E}$in$\tilde{K}$. We prove that for almost all$\unicode[STIX]{x1D70E}\in \text{Gal}(K)$, there exist infinitely many prime numbers$l$such that$A$has a nonzero$\tilde{K}(\unicode[STIX]{x1D70E})$-rational point of order$l$. This completes the proof of a conjecture of Geyer–Jarden from 1978 in characteristic 0.

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 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 $\overline{\unicode[STIX]{x1D70C}}$.

scholarly journals The distribution of the residues of a quartic polynomial

1967 ◽  
Vol 8 (2) ◽  
pp. 67-88 ◽  
Author(s):  
K. McCann ◽  
K. S. Williams
Keyword(s):  
Finite Field ◽  
Finite Number ◽  
Galois Group ◽  
Riemann Hypothesis ◽  
Function Fields ◽  
Algebraic Function ◽  
Algebraic Closure ◽  
Quartic Polynomial ◽  
Algebraic Function Fields ◽  
Image Position

Let f(x) denote a polynomial of degree d defined over a finite field k with q = pnelements. B. J. Birch and H. P. F. Swinnerton-Dyer [1] have estimated the number N(f) of distinct values of y in k for which at least one of the roots ofis in k. They prove, using A. Weil's deep results [12] (that is, results depending on the Riemann hypothesis for algebraic function fields over a finite field) on the number of points on a finite number of curves, thatwhere λ is a certain constant and the constant implied by the O-symbol depends only on d. In fact, if G(f) denotes the Galois group of the equation (1.1) over k(y) and G+(f) its Galois group over k+(y), where k+ is the algebraic closure of k, then it is shown that λ depends only on G(f), G+(f) and d. It is pointed out that “in general”

scholarly journals Extensions of number fields defined by cohomology groups

1983 ◽  
Vol 92 ◽  
pp. 179-186 ◽  
Author(s):  
Hans Opolka
Keyword(s):  
Natural Number ◽  
Galois Group ◽  
Cohomology Group ◽  
Algebraic Closure ◽  
Number Fields ◽  
Cohomology Groups ◽  
Absolute Galois Group ◽  
The Absolute

Letkbe a field of characteristic 0, letbe an algebraic closure ofkand denote byGk= G(/k) the absolute Galois group ofk. Suppose that for some natural numbern≥ 3 the cohomology groupHn(Gk) Z) is trivial.

scholarly journals THE BREUIL–MÉZARD CONJECTURE FOR POTENTIALLY BARSOTTI–TATE REPRESENTATIONS

2014 ◽  
Vol 2 ◽  
Author(s):  
TOBY GEE ◽  
MARK KISIN
Keyword(s):  
Galois Group ◽  
Finite Extension ◽  
Two Dimensional ◽  
Absolute Galois Group ◽  
Serre's Conjecture ◽  
Weight Part ◽  
The Absolute ◽  
Serre’S Conjecture

Abstract We prove the Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate representations of the absolute Galois group $G_{K}$ , $K$ a finite extension of $\mathbb{Q}_{p}$ , for any $p>2$ (up to the question of determining precise values for the multiplicities that occur). In the case that $K/\mathbb{Q}_{p}$ is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serre’s conjecture, proving a variety of results including the Buzzard–Diamond–Jarvis conjecture.

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