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 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 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.

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 Crystalline extensions and the weight part of Serre’s conjecture

2012 ◽  
Vol 6 (7) ◽  
pp. 1537-1559 ◽  
Author(s):  
Toby Gee ◽  
Tong Liu ◽  
David Savitt
Keyword(s):  
Serre's Conjecture ◽  
Weight Part ◽  
Serre’S Conjecture

scholarly journals Modularity of nearly ordinary 2-adic residually dihedral Galois representations

2014 ◽  
Vol 150 (8) ◽  
pp. 1235-1346 ◽  
Author(s):  
Patrick B. Allen
Keyword(s):  
Elliptic Curves ◽  
Galois Representations ◽  
Real Field ◽  
Two Dimensional ◽  
Hida Families ◽  
Totally Real ◽  
Totally Real Fields ◽  
Totally Real Field

AbstractWe prove modularity of some two-dimensional,$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$-adic Galois representations over a totally real field that are nearly ordinary at all places above$2$and that are residually dihedral. We do this by employing the strategy of Skinner and Wiles, using Hida families, together with the$2$-adic patching method of Khare and Wintenberger. As an application we deduce modularity of some elliptic curves over totally real fields that have good ordinary or multiplicative reduction at places above $2$.

Serre’s conjecture and its consequences

2010 ◽  
Vol 5 (1) ◽  
pp. 103-125 ◽  
Author(s):  
Chandrashekhar Khare
Keyword(s):  
Serre's Conjecture ◽  
Serre’S Conjecture
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