Modularity of the Consani-Scholten quintic

We prove that the Consani-Scholten quintic, a Calabi-Yau threefold over QQ, is Hilbert modular. For this, we refine several techniques known from the context of modular forms. Most notably, we extend the Faltings-Serre-Livne method to induced four-dimensional Galois representations over QQ. We also need a Sturm bound for Hilbert modular forms; this is developed in an appendix by Jose Burgos Gil and the second author.Comment: 35 pages, one figure; with an appendix by Jose Burgos Gil and Ariel Pacetti; v3: corrections and improvements thanks to the refere

Even Galois Representations and the Fontaine--Mazur conjecture II

We prove, under mild hypotheses, that there are no irreducible two-dimensional_even_ Galois representations of \Gal(\Qbar/\Q) which are de Rham with distinct Hodge--Tate weights. This removes the "ordinary" hypothesis required in previous work of the author. We construct examples of irreducible two-dimensional residual representations that have no characteristic zero geometric (= de Rham) deformations.Comment: Updated to take into account suggestions of the referee; the main theorems remain unchange

p-adic Hodge-theoretic properties of \'etale cohomology with mod p coefficients, and the cohomology of Shimura varieties

We show that the mod p cohomology of a smooth projective variety with semistable reduction over K, a finite extension of Qp, embeds into the reduction modulo p of a semistable Galois representation with Hodge-Tate weights in the expected range (at least after semisimplifying, in the case of the cohomological degree > 1). We prove refinements with descent data, and we apply these results to the cohomology of unitary Shimura varieties, deducing vanishing results and applications to the weight part of Serre's conjecture.Comment: Essentially final version; to appear in Algebra and Number Theor