We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold forétale cohomology with nontrivial coefficients, as well as in the relative setting, i.e. for proper smooth morphisms of smooth formal schemes. The proof is formulated in terms of the pro-étale topos introduced by Scholze, and uses his primitive comparison theorem for the structure sheaf on the proetale site. Moreover, we need to prove the Poincaré lemma for crystalline period sheaves, for which we adapt the idea of Andreatta and Iovita. Another ingredient for the proof is the geometric acyclicity of crystalline period sheaves, whose computation is due to Andreatta and Brinon.

Abstract. We prove the Breuil-Mézard conjecture for split non-scalar residual representations of Gal(Q p /Qp) by local methods. Combined with the cases previously proved in [20] and [26], this completes the proof of the conjecture (when p ≥ 5). As a consequence, the local restriction in the proof of the Fontaine-Mazur conjecture in [20] is removed. Contents Notation• p ≥ 5 is a prime number. The p-adic valuation is normalized as v p (p) = 1.• E/Q p is a sufficiently large finite extension with ring of integers O, a (fixed) uniformizer ̟, and residue field F. Its subring of Witt vectors is denoted by W (F).• For a number field F , the completion at a place v is written as F v , for which we fix a uniformizer denoted by ̟ v . • ½ : G Qp → F × p is the trivial character. We also let ½ denote other trivial representations, if no confusion arises.• Normalize the local class field map Q × p → G ab Qp so that uniformizers correspond to geometric Frobenii. Then a character of G Qp will also be regarded as a character of Q × p .• For a ring R, m-SpecR denotes the set of maximal ideals.• For R a noetherian ring and M a finite R-module of dimension at most d, let ℓ Rp (M p ) denote the length of the R p -module M p , and letWhen the context is clear, we simply denote it by Z(M ).• For R a noetherian local ring with maximal ideal m and M a finite Rmodule, and for an m-primary ideal q of R, let e q (R, M ) denote the HilbertSamuel multiplicity of M with respect to q. We abbreviate e m (R, M ) = e(R, M ) and e q (R, R) = e q (R).• For r ≥ 0, we let Sym r E 2 (resp. Sym r F 2 ) be the usual symmetric power representation of GL 2 (Z p ) (resp. of GL 2 (F p ), but viewed as a representation of GL 2 (Z p )).

Abstract.We construct one-parameter families of overconvergent Siegel-Hilbert modular forms. This result has applications to construction of Galois representations for automorphic forms of non-cohomological weights.

We study the modularity problem of Calabi-Yau varieties from the conformal field theoretic point of view. We express the modular forms associated to all 1-dimensional Calabi-Yau orbifolds in terms of products of Dedekind eta functions, which is hoped to shed light on the modularity questions for higher dimensional Calabi-Yau varieties.

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