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

Abstract

In 1987 Serre conjectured that any mod l ("ell", not "1") two-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation 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 Vigneras, 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 show how it implies the conjecture.

Version 5: Our Shimura variety conventions differ from Emerton's (see footnote 4) and so (a) some chi's needed to be changed to chi^{-1}'s in section 4 and (b) our Eichler-Shimura relation needed fixing too (also in section 4).

References in corpus (1)

• The weight in a Serre-type conjecture for tame n-dimensional Galois representations

Cited by in corpus (37)

• The weight in a Serre-type conjecture for tame n-dimensional Galois representations
• Formes modulaires de Hilbert modulo p et valeurs d'extensions galoisiennes
• G-valued local deformation rings and global lifts
• Weight cycling and Serre-type conjectures for unitary groups
• Ordinary representations of G(Q_p) and fundamental algebraic representations
• Completed cohomology of Shimura curves and a p-adic Jacquet-Langlands correspondence
• Patching and the p-adic Langlands program for GL(2, Q_p)
• Diagrams in the mod $p$ cohomology of Shimura curves
• Serre weights for quaternion algebras
• A Serre weight conjecture for geometric Hilbert modular forms in characteristic p
• Weight elimination in Serre-type conjectures
• Multiplicity one at full congruence level
• Ihara's lemma for Shimura curves over totally real fields via patching
• Multiplicity one for wildly ramified representations
• Unramifiedness of Galois representations attached to weight one Hilbert modular eigenforms mod p
• The weight reduction of mod $p$ Siegel modular forms for $GS{p}_4$
• On Serre's modularity conjecture and Fermat's equation over quadratic imaginary fields of class number one
• Extensions of rank one (phi, Gamma)-modules and crystalline representations
• Explicit Serre weights for two-dimensional Galois representations
• Serre weights and wild ramification in two-dimensional Galois representations
• Companion Forms in Parallel Weight One
• On the universal mod $p$ supersingular quotients for ${\mathrm{GL}}_2(F)$ over ${\overline{\mathbb{F}}}_p$ for a general $F/{\mathbb{Q}}_p$
• The cone of minimal weights for mod $p$ Hilbert modular forms
• Patching and Multiplicity $2^k$ for Shimura Curves
• Serre weights for locally reducible two-dimensional Galois representations
• Conjectures and results on modular representations of ${\mathrm{GL}}_n(K)$ for a $p$-adic field $K$
• Nonsolvable number fields ramified only at 3 and 5
• Serre weight conjectures for $p$-adic unitary groups of rank 2
• Finite descent obstruction for Hilbert modular varieties
• On mod p non-abelian Lubin-Tate theory for GL_2(Q_p)
• The geometric Breuil-Mézard conjecture for two-dimensional potentially Barsotti-Tate Galois representations
• Crystalline liftings and weight part of Serre's conjecture
• Modularity of ${\mathrm{PGL}}_2({\mathbb{F}}_p)$-representations over totally real fields
• Sur quelques représentations supersingulières de $G{L}_2(Q_{p^f})$
• Serre's Modularity Conjecture
• On the constituents of the mod $p$ cohomology of Shimura curves
• Globally realizable components of local deformation rings