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.

scholarly journals A geometric perspective on the Breuil–Mézard conjecture

2013 ◽  
Vol 13 (1) ◽  
pp. 183-223 ◽  
Author(s):  
Matthew Emerton ◽  
Toby Gee
Keyword(s):  
Galois Group ◽  
Type Theorem ◽  
Finite Extension ◽  
Unitary Groups ◽  
Strong Version ◽  
Two Dimensional ◽  
Absolute Galois Group ◽  
Weight Part ◽  
The Absolute ◽  
Formula Type

AbstractLet$p\gt 2$be prime. We state and prove (under mild hypotheses on the residual representation) a geometric refinement of the Breuil–Mézard conjecture for two-dimensional mod$p$representations of the absolute Galois group of${ \mathbb{Q} }_{p} $. We also state a conjectural generalization to$n$-dimensional representations of the absolute Galois group of an arbitrary finite extension of${ \mathbb{Q} }_{p} $, and give a conditional proof of this conjecture, subject to a certain$R= \mathbb{T} $-type theorem together with a strong version of the weight part of Serre’s conjecture for rank $n$unitary groups. We deduce an unconditional result in the case of two-dimensional potentially Barsotti–Tate 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 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.

scholarly journals THE WEIGHT PART OF SERRE’S CONJECTURE FOR

2015 ◽  
Vol 3 ◽  
Author(s):  
TOBY GEE ◽  
TONG LIU ◽  
DAVID SAVITT
Keyword(s):  
Two Dimensional ◽  
Crystalline Representations ◽  
Serre's Conjecture ◽  
Weight Part ◽  
Local Methods ◽  
Totally Real ◽  
Totally Real Fields ◽  
Serre’S Conjecture

AbstractLet $p>2$ be prime. We use purely local methods to determine the possible reductions of certain two-dimensional crystalline representations, which we call pseudo-Barsotti–Tate representations, over arbitrary finite extensions of $\mathbb{Q}_{p}$. As a consequence, we establish (under the usual Taylor–Wiles hypothesis) the weight part of Serre’s conjecture for $\text{GL}(2)$ over arbitrary totally real fields.

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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