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.

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In this paper, we prove that, to any Hubert cuspidal eigenform, one may attach a compatible System of Galois representations. This result extends the analogous results of Deligne and Deligne-Serre for elliptic modular forms. The principal work on this conjecture was carried out by Carayol and Taylor, but their results left one case remaining, which we complete in this paper. We also investigate the compatibility of our results with the local Langlands correspondence, and prove that whenever the local component of the automorphic representation is not special, then the results coincide.I should like to take this opportunity to thank Richard Taylor for suggesting this problem, and for bis help during the time that the work on this paper was carried out.

In this short note, we develop the Stienstra-Beukers theory of supercongruences in the setting of the Catalan-Larcombe-French sequence. We also give some applications to other sequences.

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