We derive a formula for the Chern classes of the bundles of conformal blocks on M 0,n associated to simple finite dimensional Lie algebras and explore its conseqences in more detail for g = sl 2 and for arbitrary g and level 1. We also give a method for computing the first Chern class of such bundles on M g,n for g > 0.

We study irreducible odd mod p Galois representations ρ : Gal(F/F) → G(F p ), for F a totally real number field and G a general reductive group. For p ≫ G,F 0, we show that any ρ that lifts locally, and at places above p to de Rham and Hodge-Tate regular representations, has a geometric p-adic lift. We also prove non-geometric lifting results without any oddness assumption.

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Abstract. We compute the rational Chow groups of supersingular abelian varieties and some other related varieties, such as supersingular Fermat varieties and supersingular K3 surfaces. These computations are concordant with the conjectural relationship, for a smooth projective variety, between the structure of Chow groups and the coniveau filtration on the cohomology.

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