Work

On Serre's conjecture for mod ℓ Galois representations over totally real fields

Year: 2010

Type: article

Abstract: In 1987 Serre conjectured that any mod ℓ 2-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalization of this conjecture to 2-dimensional representations of the absolute Galois group of a totally real field where ℓ is unramified. The hard work is in formulating an analogue of the weight part of Serre's conjecture. Serre furthermore asked whether his conjecture could be rephrased in terms of a “mod ℓ Langlands philosophy.” Using ideas of Emerton and Vignéras, we formulate a mod ℓ local-global principle for the group D*, where D is a quaternion algebra over a totally real field, split above ℓ and at 0 or 1 infinite places, and we show how it implies the conjecture.

Source: Duke Mathematical Journal

DOI: 10.1215/00127094-2010-052

Authors: Kevin Buzzard, Fred Diamond, Frazer Jarvis

Institutions: Imperial College London, King's College London, University of Sheffield

Language: English

Topic: Advanced Algebra and Geometry

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, American Institute of Mathematics, Engineering and Physical Sciences Research Council

OpenAlex ID: W2132228041

FWCI: 19.23

Cites: 68

Cited by: 126

Related to: 10