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 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.

On arithmetic families of filtered -modules and crystalline representations

2012 ◽  
Vol 12 (4) ◽  
pp. 677-726 ◽  
Author(s):  
Eugen Hellmann
Keyword(s):  
Galois Group ◽  
Galois Representations ◽  
Absolute Galois Group ◽  
Varying Coefficients ◽  
Crystalline Representations ◽  
The Absolute ◽  
Integral Data ◽  
Filtered Modules

AbstractWe consider stacks of filtered$\varphi $-modules over rigid analytic spaces and adic spaces. We show that these modules parameterize$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 parameterizing integral data, and determine the image of this morphism.

scholarly journals The semisimplicity conjecture for A-motives

2010 ◽  
Vol 146 (3) ◽  
pp. 561-598 ◽  
Author(s):  
Nicolas Stalder
Keyword(s):  
Automorphism Group ◽  
Galois Group ◽  
Zariski Closure ◽  
Drinfeld Modules ◽  
Absolute Galois Group ◽  
Connected Component ◽  
Finitely Generated ◽  
Endomorphism Algebra ◽  
Monodromy Groups ◽  
The Absolute

AbstractWe prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V𝔭(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G𝔭(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G𝔭(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V𝔭(M). The second states that the connected component of G𝔭(M) is reductive if M is semisimple and has a separable endomorphism algebra.

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.

scholarly journals The Sylow subgroups of the absolute Galois group Gal(Q)

2015 ◽  
Vol 284 ◽  
pp. 186-212 ◽  
Author(s):  
Lior Bary-Soroker ◽  
Moshe Jarden ◽  
Danny Neftin
Keyword(s):  
Galois Group ◽  
Absolute Galois Group ◽  
Sylow Subgroups ◽  
The Absolute

Zeta functions of twisted modular curves

2006 ◽  
Vol 80 (1) ◽  
pp. 89-103 ◽  
Author(s):  
Cristian Virdol
Keyword(s):  
Zeta Function ◽  
Galois Group ◽  
Complex Plane ◽  
Zeta Functions ◽  
Modular Curves ◽  
Absolute Galois Group ◽  
Modular Curve ◽  
The Absolute

AbstractIn this paper we compute and continue meromorphically to the whole complex plane the zeta function for twisted modular curves. The twist of the modular curve is done by a modprepresentation of the absolute Galois group.

scholarly journals On L-Functions of Twisted 3-Dimensional Quaternionic Shimura Varieties

2008 ◽  
Vol 190 ◽  
pp. 87-104
Author(s):  
Cristian Virdol
Keyword(s):  
Galois Group ◽  
Complex Plane ◽  
Zeta Functions ◽  
Shimura Varieties ◽  
Absolute Galois Group ◽  
3 Dimensional ◽  
Entire Complex ◽  
The Absolute

In this paper we compute and continue meromorphically to the entire complex plane the zeta functions of twisted quaternionic Shimura varieties of dimension 3. The twist of the quaternionic Shimura varieties is done by a mod ℘ representation of the absolute Galois group.

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