Serre weights and the Breuil-Mézard conjecture for modular forms

潜在Barsotti-Tate表示的Breuil-Mézard猜想

The Breuil-Mézard conjecture for potentially Barsotti-Tate representations

本文证明了对于p>2，绝对Galois群G_K的二维潜在Barsotti-Tate表示的Breuil-Mézard猜想，并应用于Serre猜想的权部分，包括Buzzard-Diamond-Jarvis猜想。

Potentially crystalline deformation rings and Serre weight conjectures

We prove the weight part of Serre's conjecture in generic situations for forms of $U(3)$ which are compact at infinity and split at places dividing $p$ as conjectured by Herzig. We also prove automorphy lifting theorems in dimension three. The key input is an explicit description of tamely potentially crystalline deformation rings with Hodge-Tate weights $(2,1,0)$ for $K/\mathbb{Q}_p$ unramified combined with patching techniques. Our results show that the (geometric) Breuil-Mézard conjectures hold for these deformation rings.

Weight cycling and Serre-type conjectures for unitary groups

We prove that for forms of U(3) which are compact at infinity and split at places dividing a prime p, in generic situations the Serre weights of a mod p modular Galois representation which is irreducible when restricted to each decomposition group above p are exactly those previously predicted by the third author. We do this by combining explicit computations in p-adic Hodge theory, based on a formalism of strongly divisible modules and Breuil modules with descent data which we develop in the paper, with a technique that we call "weight cycling".

General Serre weight conjectures

We formulate a number of related generalisations of the weight part of Serre's conjecture to the case of GL(n) over an arbitrary number field, motivated by the formalism of the Breuil-Mézard conjecture. We give evidence for these conjectures, and discuss their relationship to previous work. We generalise one of these conjectures to the case of connected reductive groups which are unramified over Q_p, and we also generalise the second author's previous conjecture for GL(n)/Q to this setting, and show that the two conjectures are generically in agreement.

On Serre's reciprocity conjecture for 2-dimensional mod p representations of the Galois group of Q

We first prove the existence of minimally ramified p-adic lifts of 2-dimensional mod p representations, that are odd and irreducible, of the absolute Galois group of Q,in many cases. This is predicted by Serre's conjecture that such representations arise from newforms of optimal level and weight. Using these minimal lifts, and arguments using compatible systems, we prove some cases of Serre's conjectures in low levels and weights. For instance we prove that there are no irreducible (p,p) type group schemes over the rational integers. We prove that a Galois representation as above of Artin cond…

Schémas en groupes et poids de Diamond-Serre

This note is a correction of (statement and proof of) proposition 3.3.1 of Toby Gee's preprint intitled *On the weights of mod p Hilbert modular forms*. The aim is to compare Galois representations arising from extensions of some group schemes (over the ring of integers of a p-adic field) endowed with a descent data, and extensions of some crystalline representations with given Hodge-Tate weights. The main tool of the proof is the theory of Breuil.

中心除代数上的 Breuil-Mézard 猜想

Breuil-Mézard conjectures for central division algebras

本文将 Breuil-Mézard 猜想推广到中心除代数的单位群，利用 Deligne-Lusztig 理论构造惯性类型与 Serre 权重的转移，并借助惯性 Jacquet-Langlands 对应和显式特征标公式证明其可由 GL_n 情形推出。

Serre weights, Galois deformation rings, and local models

We survey some recent progress on generalizations of conjectures of Serre concerning the cohomology of arithmetic groups, focusing primarily on the "weight" aspect. This is intimately related to (generalizations of) a conjecture of Breuil and Mézard relating the geometry of potentially semistable deformation rings to modular representation theory. Recently, B. Levin, S. Morra, and the authors established these conjectures in tame generic contexts by constructing projective varieties (local models) in mixed characteristic whose singularities model, in generic cases, those of tamely potentially …

三维强分歧 Galois 表示的 Serre 权

Serre weights for three-dimensional wildly ramified Galois representations

本文证明三维 mod p Galois 表示在 p 不分歧且满足一般性条件时的 Serre 权猜想，去除弱分歧假设，并建立 Breuil 格猜想与 U(3) 算术流形重数为一结果。

晶体现象空间的解析与模性

Resolutions of spaces of crystalline representations and modularity

本文研究晶体现象空间的解析，证明在Hodge-Tate权重间隙小于p时，部分解析的存在性，并在最小正则权重情况下展示解析的正常性，从而推导出自动性提升、塞雷猜想的权重部分及三维最小正则权重下的布里乌-梅兹ard猜想。

Local models for Galois deformation rings and applications

We construct projective varieties in mixed characteristic whose singularities model, in generic cases, those of tamely potentially crystalline Galois deformation rings for unramified extensions of $\mathbb{Q}_p$ with small regular Hodge-Tate weights. We establish several significant facts about their geometry including a unibranch property at special points and a representation theoretic description of the irreducible components of their special fibers. We derive from these geometric results a number of local and global consequences: the Breuil-Mézard conjecture in arbitrary dimension for tame…

通过μ_p-等变性对晶体伽罗瓦表示进行模p约化

Reduction modulo p of crystalline Galois representations via μ_p-equivariance

研究\(\mathbb{Q}_p\)绝对伽罗瓦群晶体表示模p约化的惯性权重，通过棱柱技术使布雷伊尔 - 基辛模模p约化获\(\mu_p\)-等变结构，结合仿射格拉斯曼流形结果，明确塞尔权重猜想并证其消除方向。