scholarly journals On mod local-global compatibility for in the ordinary case

2017 ◽  
Vol 153 (11) ◽  
pp. 2215-2286 ◽  
Author(s):  
Florian Herzig ◽  
Daniel Le ◽  
Stefano Morra
Keyword(s):  
Unitary Group ◽  
Automorphic Forms ◽  
Galois Representations ◽  
Galois Representation ◽  
Number Fields ◽  
Serre's Conjecture ◽  
Ordinary Case ◽  
Serre’S Conjecture ◽  
The Way

Suppose that $F/F^{+}$ is a CM extension of number fields in which the prime $p$ splits completely and every other prime is unramified. Fix a place $w|p$ of $F$. Suppose that $\overline{r}:\operatorname{Gal}(\overline{F}/F)\rightarrow \text{GL}_{3}(\overline{\mathbb{F}}_{p})$ is a continuous irreducible Galois representation such that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ is upper-triangular, maximally non-split, and generic. If $\overline{r}$ is automorphic, and some suitable technical conditions hold, we show that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ can be recovered from the $\text{GL}_{3}(F_{w})$-action on a space of mod $p$ automorphic forms on a compact unitary group. On the way we prove results about weights in Serre’s conjecture for $\overline{r}$, show the existence of an ordinary lifting of $\overline{r}$, and prove the freeness of certain Taylor–Wiles patched modules in this context. We also show the existence of many Galois representations $\overline{r}$ to which our main theorem applies.

Serre weights for U ⁢ ( n ) {U(n)}

2018 ◽  
Vol 2018 (735) ◽  
pp. 199-224 ◽  
Author(s):  
Thomas Barnet-Lamb ◽  
Toby Gee ◽  
David Geraghty
Keyword(s):  
Galois Representations ◽  
Galois Representation ◽  
Unitary Groups ◽  
Automorphic Representations ◽  
Serre's Conjecture ◽  
Weight Part ◽  
Serre’S Conjecture

Abstract We study the weight part of (a generalisation of) Serre’s conjecture for mod l Galois representations associated to automorphic representations on unitary groups of rank n for odd primes l. Given a modular Galois representation, we use automorphy lifting theorems to prove that it is modular in many other weights. We make no assumptions on the ramification or inertial degrees of l. We give an explicit strengthened result when {n=3} and l splits completely in the underlying CM field.

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.

scholarly journals SERRE WEIGHTS FOR LOCALLY REDUCIBLE TWO-DIMENSIONAL GALOIS REPRESENTATIONS

2014 ◽  
Vol 14 (3) ◽  
pp. 639-672 ◽  
Author(s):  
Fred Diamond ◽  
David Savitt
Keyword(s):  
Galois Representations ◽  
Real Field ◽  
Two Dimensional ◽  
Serre's Conjecture ◽  
Weight Part ◽  
Totally Real ◽  
Serre’S Conjecture ◽  
Totally Real Field

Let $F$ be a totally real field, and $v$ a place of $F$ dividing an odd prime $p$. We study the weight part of Serre’s conjecture for continuous totally odd representations $\overline{{\it\rho}}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbb{F}}_{p})$ that are reducible locally at $v$. Let $W$ be the set of predicted Serre weights for the semisimplification of $\overline{{\it\rho}}|_{G_{F_{v}}}$. We prove that, when $\overline{{\it\rho}}|_{G_{F_{v}}}$ is generic, the Serre weights in $W$ for which $\overline{{\it\rho}}$ is modular are exactly the ones that are predicted (assuming that $\overline{{\it\rho}}$ is modular). We also determine precisely which subsets of $W$ arise as predicted weights when $\overline{{\it\rho}}|_{G_{F_{v}}}$ varies with fixed generic semisimplification.

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.

DEFORMATIONS OF GALOIS REPRESENTATIONS AND THE THEOREMS OF SATO–TATE AND LANG–TROTTER

2011 ◽  
Vol 07 (08) ◽  
pp. 2065-2079
Author(s):  
AFTAB PANDE
Keyword(s):  
Galois Representations ◽  
Galois Representation ◽  
Number Fields

We construct infinitely ramified Galois representations ρ such that the al(ρ)'s have distributions in contrast to the statements of Sato–Tate, Lang–Trotter and others. Using similar methods we deform a residual Galois representation for number fields and obtain an infinitely ramified representation with very large image, generalizing a result of Ramakrishna.

THE NONEXISTENCE OF CERTAIN NONSOLVABLE GALOIS EXTENSIONS OF NUMBER FIELDS OF SMALL DEGREE

2005 ◽  
Vol 01 (01) ◽  
pp. 155-160 ◽  
Author(s):  
SHARON BRUEGGEMAN
Keyword(s):  
Number Fields ◽  
Small Degree ◽  
Galois Extensions ◽  
Serre's Conjecture ◽  
Serre’S Conjecture

Serre's conjecture predicts the nonexistence of certain nonsolvable Galois extensions of ℚ which are unramified outside one small prime. These nonexistence theorems have been proven by the techniques of discriminant bounding. In this paper, we will apply these techniques to nonsolvable extensions of small degree number fields.

scholarly journals Mod pq Galois representations and Serre's conjecture

2003 ◽  
Vol 98 (2) ◽  
pp. 329-347 ◽  
Author(s):  
Chandrashekhar Khare ◽  
Ian Kiming
Keyword(s):  
Galois Representations ◽  
Serre's Conjecture ◽  
Serre’S Conjecture

scholarly journals Images of Galois representations in mod p Hecke algebras

2020 ◽  
pp. 1-21
Author(s):  
Laia Amorós
Keyword(s):  
Finite Field ◽  
Commutative Algebra ◽  
Galois Representations ◽  
Residue Field ◽  
Representation Formula ◽  
Galois Representation ◽  
Number Fields ◽  
Abelian Extensions ◽  
Hecke Eigenform ◽  
Coefficient Ring

Let [Formula: see text] denote the mod [Formula: see text] local Hecke algebra attached to a normalized Hecke eigenform [Formula: see text], which is a commutative algebra over some finite field [Formula: see text] of characteristic [Formula: see text] and with residue field [Formula: see text]. By a result of Carayol we know that, if the residual Galois representation [Formula: see text] is absolutely irreducible, then one can attach to this algebra a Galois representation [Formula: see text] that is a lift of [Formula: see text]. We will show how one can determine the image of [Formula: see text] under the assumptions that (i) the image of the residual representation contains [Formula: see text], (ii) [Formula: see text] and (iii) the coefficient ring is generated by the traces. As an application we will see that the methods that we use allow to deduce the existence of certain [Formula: see text]-elementary abelian extensions of big non-solvable number fields.

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