General Serre weight conjectures (1509.02527v3)

Abstract: 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-M\'ezard 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.

General Serre Weight Conjectures: An Overview

Introduction

The research paper titled "General Serre Weight Conjectures" by Toby Gee, Florian Herzig, and David Savitt explores the formulation and generalization of the weight part of Serre's conjecture—a pivotal topic within algebraic geometry and number theory. Originally proposed by Jean-Pierre Serre, this conjecture concerns the modularity of an odd irreducible representation $r: G_{\mathbb{Q}} \to GL_2(\mathbb{F}_p)$ under specific conditions on level and weight. The presented paper extends these conjectural frameworks to $GL_n$ over arbitrary number fields and provides evidence supporting these generalized conjectures.

Serre's Conjecture for Modular Forms

Serre's conjecture focuses on establishing connections between Galois representations and modular forms, wherein a Galois representation $r$ arises from a modular form of specific properties—particularly influencing the modularity level and weight. This connection is essential not only for understanding number theoretic transformations but also for broader implications in the Langlands program. The authors meticulously generalize these weight predictions to the context of $GL_n$ over arbitrary number fields, offering new insights and conjectural models applicable to connected reductive groups unramified over $\mathbb{Q}_p$.

Framework and Proposed Generalizations

The paper outlines several related conjectures and discusses their motivation alongside the development of necessary mathematical structures. Key among these is the Breuil-Mézard conjecture, which ties the geometry of deformation spaces of local Galois representations to the modular representation theory of $GL_2$. This conjecture helps illuminate the local-global compatibility within modular forms and Galois theory, positing new pathways for understanding deformation theory's role in broader number-theoretic contexts.

Furthermore, the authors explore early generalizations including works by Buzzard, Diamond, and Jarvis, emphasizing the significance of Serre weights in this context. This approach emphasizes that understanding modular forms through geometric points of view opens novel avenues for prediction and computational verification in number theory.

Analysis and Rationale

The authors employ intricate lattice structures, deformation rings, and Hodge types to offer conjectures regarding Galois representation weight sets. The conjectures are substantiated by leveraging both patching functors and the broader Taylor–Wiles–Kisin method, providing a cohesive rationale for linking $GL_n$-representational modularity with algebraic geometry. In doing so, they also establish connections with earlier validated conjectures and offer computational examples that align with their predictions, maintaining a strong evidentiary basis and extending methodologies from $GL_2$ to arbitrary dimensions.

Implications and Future Directions

This paper provides robust implications for extending Serre's conjecture beyond traditional field boundaries, enabling a deeper understanding of the relationships between modular forms and Galois representations within new mathematical frameworks. These implications are influential for both theoretical development and practical computational applications in number theory.

The authors speculate that future developments may include further explorations into modularity lifting theorems and potential advancements in the mod $p$ Langlands program. The resolution of weight conjectures in more complex number fields and the reconciliation of divergent geometric concepts remain as prospective studies following this research. Overall, the paper delineates a clear pathway toward integrated conjectural frameworks, fostering broader insights and potential breakthroughs in advanced number theory and algebraic geometry.