Abstract. In 1987 Serre conjectured that any mod ℓ 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 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 show how it implies the conjecture.

No abstract

No abstract

We prove that, near the boundary of weight space, the 2-adic eigencurve of tame level 1 can be written as an infinite disjoint union of 'evenly spaced' annuli, and on each annulus the slopes of the corresponding overconvergent eigenforms tend to zero.

No abstract