scholarly journals Explicit Computation of a Galois Representation Attached to an Eigenform Over $${\text {SL}}_3$$ from the $${{\text {H}}}_{\acute{\mathrm{e}}\mathrm{t}}^2$$ of a Surface

2022 ◽  
Author(s):  
Nicolas Mascot
Keyword(s):  
Simple Group ◽  
Galois Group ◽  
Congruence Subgroup ◽  
Galois Representations ◽  
Galois Representation ◽  
Explicit Computation

AbstractWe describe a method to compute mod $$\ell $$ ℓ Galois representations contained in the $${{\text {H}}}_{\acute{\mathrm{e}}\mathrm{t}}^2$$ H e ´ t 2 of surfaces. We apply this method to the case of a representation with values in $${\text {GL}}_3(\mathbb {F}_9)$$ GL 3 ( F 9 ) attached to an eigenform over a congruence subgroup of $${\text {SL}}_3$$ SL 3 . We obtain, in particular, a polynomial with Galois group isomorphic to the simple group $${\text {PSU}}_3(\mathbb {F}_9)$$ PSU 3 ( F 9 ) and ramified at 2 and 3 only.

scholarly journals Big Galois representations and -adic -functions

2014 ◽  
Vol 151 (4) ◽  
pp. 603-664 ◽  
Author(s):  
Haruzo Hida
Keyword(s):  
Power Series ◽  
Irreducible Component ◽  
Principal Ideal ◽  
Congruence Subgroup ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Principal Congruence ◽  
Principal Congruence Subgroup ◽  
Canonical Weight

Let$p\geqslant 5$be a prime. If an irreducible component of the spectrum of the ‘big’ ordinary Hecke algebra does not have complex multiplication, under mild assumptions, we prove that the image of its Galois representation contains, up to finite error, a principal congruence subgroup${\rm\Gamma}(L)$of$\text{SL}_{2}(\mathbb{Z}_{p}[[T]])$for a principal ideal$(L)\neq 0$of$\mathbb{Z}_{p}[[T]]$for the canonical ‘weight’ variable$t=1+T$. If$L\notin {\rm\Lambda}^{\times }$, the power series$L$is proven to be a factor of the Kubota–Leopoldt$p$-adic$L$-function or of the square of the anticyclotomic Katz$p$-adic$L$-function or a power of$(t^{p^{m}}-1)$.

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 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.

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

AbstractLetlbe a prime number. In this paper, we prove that the isomorphism class of anl-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-louter 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.

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.

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