Crystalline liftings and the weight part of Serre’s conjecture

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.

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THE WEIGHT PART OF SERRE’S CONJECTURE FOR

Forum of Mathematics Pi ◽

10.1017/fmp.2015.1 ◽

2015 ◽

Vol 3 ◽

Cited By ~ 9

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.

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THE BREUIL–MÉZARD CONJECTURE FOR POTENTIALLY BARSOTTI–TATE REPRESENTATIONS

Forum of Mathematics Pi ◽

10.1017/fmp.2014.1 ◽

2014 ◽

Vol 2 ◽

Cited By ~ 20

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.

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Serre weights for U ⁢ ( n ) {U(n)}

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2015-0015 ◽

2018 ◽

Vol 2018 (735) ◽

pp. 199-224 ◽

Cited By ~ 3

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.

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