Kummer theory for big Galois representations

2007 ◽  
Vol 142 (2) ◽  
pp. 205-217 ◽  
Author(s):  
DANIEL DELBOURGO ◽  
PAUL SMITH
Keyword(s):  
Abelian Variety ◽  
Local Field ◽  
Galois Representations ◽  
Galois Representation ◽  
Kummer Theory ◽  
Local Pairing

AbstractIn their 1990 paper, Bloch and Kato described the image of the Kummer map on an abelian variety over a local field, as the group of 1-cocycles which trivialise after tensoring by Fontaine's mysterious ring BdR. We prove the analogue of this statement for the universal nearly-ordinary Galois representation. The proof uses a generalisation of the Tate local pairing to representations over affinoid K-algebras.

Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves

2014 ◽  
Vol 66 (5) ◽  
pp. 1167-1200 ◽  
Author(s):  
Victor Rotger ◽  
Carlos de Vera-Piquero
Keyword(s):  
Abelian Variety ◽  
Moduli Space ◽  
Quadratic Field ◽  
Galois Representations ◽  
Abelian Varieties ◽  
Main Idea ◽  
Galois Representation ◽  
Rational Points ◽  
Shimura Curves ◽  
Fields Of Moduli

AbstractThe purpose of this note is to introduce a method for proving the non-existence of rational points on a coarse moduli space X of abelian varieties over a given number field K in cases where the moduli problem is not fine and points in X(K) may not be represented by an abelian variety (with additional structure) admitting a model over the field K. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired by the work of Ellenberg and Skinner on the modularity of ℚ-curves, is that one may still attach a Galois representation of Gal(/K) with values in the quotient group GL(Tℓ(A))/ Aut(A) to a point P = [A] ∈ X(K) represented by an abelian variety A/, provided Aut(A) lies in the centre of GL(Tℓ(A)). We exemplify our method in the cases where X is a Shimura curve over an imaginary quadratic field or an Atkin–Lehner quotient over ℚ.

Computing Galois representations of modular abelian surfaces

2014 ◽  
Vol 17 (A) ◽  
pp. 36-48 ◽  
Author(s):  
Jinxiang Zeng
Keyword(s):  
Abelian Variety ◽  
Efficient Algorithm ◽  
Galois Representations ◽  
Abelian Surfaces ◽  
Genus Two

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f\in S_2(\Gamma _0(N))$ be a normalized newform such that the abelian variety $A_f$ attached by Shimura to $f$ is the Jacobian of a genus-two curve. We give an efficient algorithm for computing Galois representations associated to such newforms.

scholarly journals Nearly ordinary Galois deformations over arbitrary number fields

2008 ◽  
Vol 8 (1) ◽  
pp. 99-177 ◽  
Author(s):  
Frank Calegari ◽  
Barry Mazur
Keyword(s):  
Arbitrary Number ◽  
Quadratic Field ◽  
Galois Representations ◽  
Galois Representation ◽  
Number Fields ◽  
Base Change ◽  
Deformation Spaces ◽  
Totally Real ◽  
Set Of Points ◽  
Dense Set

AbstractLet K be an arbitrary number field, and let ρ : Gal($\math{\bar{K}}$/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation space associated to ρ contains a Zariski dense set of points corresponding to ‘automorphic’ Galois representations. We conjecture that if K is not totally real, then this is never the case, except in three exceptional cases, corresponding to: (1) ‘base change’, (2) ‘CM’ forms, and (3) ‘even’ representations. The latter case conjecturally can only occur if the image of ρ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of ‘many’ positive-dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about ‘p-adic functoriality’, as well as some remarks on how our methods should apply to n-dimensional representations of Gal($\math{\bar{\QQ}}$/ℚ) when n > 2.

On the Surjectivity of the Galois Representations Associated to Non-CM Elliptic Curves

2005 ◽  
Vol 48 (1) ◽  
pp. 16-31 ◽  
Author(s):  
Alina Carmen Cojocaru ◽  
Ernst Kani
Keyword(s):  
Positive Integer ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Upper Bounds

AbstractLet E be an elliptic curve defined over ℚ, of conductor N and without complex multiplication. For any positive integer l, let ϕl be the Galois representation associated to the l-division points of E. From a celebrated 1972 result of Serre we know that ϕl is surjective for any sufficiently large prime l. In this paper we find conditional and unconditional upper bounds in terms of N for the primes l for which ϕl is not surjective.

scholarly journals DISTINGUISHED MODELS OF INTERMEDIATE JACOBIANS

2018 ◽  
Vol 19 (3) ◽  
pp. 891-918 ◽  
Author(s):  
Jeffrey D. Achter ◽  
Sebastian Casalaina-Martin ◽  
Charles Vial
Keyword(s):  
Abelian Variety ◽  
Hilbert Scheme ◽  
Projective Variety ◽  
Galois Representation ◽  
Complete Intersections ◽  
Base Field ◽  
Field Of Definition ◽  
Level 1 ◽  
Albanese Variety ◽  
Coniveau Filtration

We show that the image of the Abel–Jacobi map admits functorially a model over the field of definition, with the property that the Abel–Jacobi map is equivariant with respect to this model. The cohomology of this abelian variety over the base field is isomorphic as a Galois representation to the deepest part of the coniveau filtration of the cohomology of the projective variety. Moreover, we show that this model over the base field is dominated by the Albanese variety of a product of components of the Hilbert scheme of the projective variety, and thus we answer a question of Mazur. We also recover a result of Deligne on complete intersections of Hodge level 1.

scholarly journals UNRAMIFIEDNESS OF GALOIS REPRESENTATIONS ATTACHED TO HILBERT MODULAR FORMS MOD  OF WEIGHT 1

2018 ◽  
Vol 19 (2) ◽  
pp. 281-306 ◽  
Author(s):  
Mladen Dimitrov ◽  
Gabor Wiese
Keyword(s):  
Modular Forms ◽  
Galois Representations ◽  
Galois Representation ◽  
Important Case ◽  
Hilbert Modular Forms ◽  
Hilbert Modular

The main result of this article states that the Galois representation attached to a Hilbert modular eigenform defined over $\overline{\mathbb{F}}_{p}$ of parallel weight 1 and level prime to $p$ is unramified above $p$. This includes the important case of eigenforms that do not lift to Hilbert modular forms in characteristic 0 of parallel weight 1. The proof is based on the observation that parallel weight 1 forms in characteristic $p$ embed into the ordinary part of parallel weight $p$ forms in two different ways per prime dividing $p$, namely via ‘partial’ Frobenius operators.

Braid monodromies on proper curves and pro-ℓ Galois representations

2011 ◽  
Vol 11 (1) ◽  
pp. 161-188 ◽  
Author(s):  
Naotake Takao
Keyword(s):  
Galois Representations ◽  
Galois Representation ◽  
Number Fields ◽  
Mapping Class ◽  
Algebraic Number Fields ◽  
Universal Family ◽  
Hyperbolic Curve ◽  
Graded Lie Algebra ◽  
The One ◽  
First Time

AbstractLetCbe a proper smooth geometrically connected hyperbolic curve over a field of characteristic 0 and ℓ a prime number. We prove the injectivity of the homomorphism from the pro-ℓ mapping class group attached to the two dimensional configuration space ofCto the one attached toC, induced by the natural projection. We also prove a certain graded Lie algebra version of this injectivity. Consequently, we show that the kernel of the outer Galois representation on the pro-ℓ pure braid group onCwithnstrings does not depend onn, even ifn= 1. This extends a previous result by Ihara–Kaneko. By applying these results to the universal family over the moduli space of curves, we solve completely Oda's problem on the independency of certain towers of (infinite) algebraic number fields, which has been studied by Ihara, Matsumoto, Nakamura, Ueno and the author. Sequentially we obtain certain information of the image of this Galois representation and get obstructions to the surjectivity of the Johnson–Morita homomorphism at each sufficiently large even degree (as Oda predicts), for the first time for a proper curve.

scholarly journals Frobenius fields for Drinfeld modules of rank 2

2008 ◽  
Vol 144 (4) ◽  
pp. 827-848 ◽  
Author(s):  
Alina Carmen Cojocaru ◽  
Chantal David
Keyword(s):  
Quadratic Field ◽  
Galois Representations ◽  
Rational Functions ◽  
Galois Representation ◽  
Field Extension ◽  
Imaginary Quadratic Field ◽  
Drinfeld Module ◽  
Rank 2 ◽  
Imaginary Field ◽  
Square Sieve

AbstractLet ϕ be a Drinfeld module of rank 2 over the field of rational functions $F=\mathbb {F}_q(T)$, with $\mathrm {End}_{\bar {F}}(\phi ) = \mathbb {F}_q[T]$. Let K be a fixed imaginary quadratic field over F and d a positive integer. For each prime $\mathfrak {p}$ of good reduction for ϕ, let $\pi _{\mathfrak {p}}(\phi )$ be a root of the characteristic polynomial of the Frobenius endomorphism of ϕ over the finite field $\mathbb {F}_q[T] / \mathfrak {p}$. Let Πϕ(K;d) be the number of primes $\mathfrak {p}$ of degree d such that the field extension $F(\pi _{\mathfrak {p}}(\phi ))$ is the fixed imaginary quadratic field K. We present upper bounds for Πϕ(K;d) obtained by two different approaches, inspired by similar ones for elliptic curves. The first approach, inspired by the work of Serre, is to consider the image of Frobenius in a mixed Galois representation associated to K and to the Drinfeld module ϕ. The second approach, inspired by the work of Cojocaru, Fouvry and Murty, is based on an application of the square sieve. The bounds obtained with the first method are better, but depend on the fixed quadratic imaginary field K. In our application of the second approach, we improve the results of Cojocaru, Murty and Fouvry by considering projective Galois representations.

scholarly journals Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero

2011 ◽  
Vol 203 ◽  
pp. 47-100 ◽  
Author(s):  
Yuichiro Hoshi
Keyword(s):  
Isomorphism Class ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Genus Zero ◽  
Isomorphism Classes ◽  
Tate Module ◽  
Hyperbolic Curve ◽  
Theoretic Characterization

AbstractLet l be a prime number. In this paper, we prove that the isomorphism class of an l-monodromically full hyperbolic curve of genus zero over a finitely generated extension of the field of rational numbers is completely determined by the kernel of the natural pro-l outer Galois representation associated to the hyperbolic curve. This result can be regarded as a genus zero analogue of a result due to Mochizuki which asserts that the isomorphism class of an elliptic curve which does not admit complex multiplication over a number field is completely determined by the kernels of the natural Galois representations on the various finite quotients of its Tate module.

scholarly journals Locally Indecomposable Galois Representations

2011 ◽  
Vol 63 (2) ◽  
pp. 277-297 ◽  
Author(s):  
Eknath Ghate ◽  
Vinayak Vatsal
Keyword(s):  
Galois Representations ◽  
Galois Representation ◽  
Cusp Forms ◽  
The Family

Abstract In a previous paper the authors showed that, under some technical conditions, the local Galois representations attached to the members of a non-CM family of ordinary cusp forms are indecomposable for all except possibly finitely many members of the family. In this paper we use deformation theoretic methods to give examples of non-CM families for which every classical member of weight at least two has a locally indecomposable Galois representation.

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