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. Let S be a Dedekind scheme with field of functions K. We show that if X K is a smooth connected proper curve of positive genus over K, then it admits a Néron model over S, i.e., a smooth separated model of finite type satisfying the usual Néron mapping property. It is given by the smooth locus of the minimal proper regular model of X K over S, as in the case of elliptic curves. When S is excellent, a similar result holds for connected smooth affine curves different from the affine line, with locally finite type Néron models.

Let O K be a complete discrete valuation ring with algebraically closed residue field of positive characteristic and field of fractions K. Let X K be a torsor under an elliptic curve A K over K, X the proper minimal regular model of X K over S := Spec(O K ), and J the identity component of the Néron model of Pic 0 XK /K . We study the canonical morphism q : Pic 0 X/S → J which extends the natural isomorphism on generic fibres. We show that q is pro-algebraic in nature with a construction that recalls Serre's work on local class field theory. Furthermore, we interpret our results in relation to Shafarevich's duality theory for torsors under abelian varieties.

This article studies curves with singularities of "coordinate axes type". As an application, we generalize a result of Deninger and Werner (Ann Sci École Norm Sup 38(4): 2005), which is proved in the case of good reduction, to the case where the curve has semi-stable reduction.

This article concerns the geometry of algebraic curves in characteristic p > 0. We study the geometric and arithmetic properties of the theta divisor associated to the vector bundle of locally exact differential forms of a curve. Among other things, we prove that, for a generic curve of genus ≥ 2, this theta divisor is always geometrically normal. We give also some results in the case where either p or the genus of the curve is small. In the last part, we apply our results on to the study of the variation of fundamental group of algebraic curves. In particular, we refine a recent result of Tamagawa on the specialization homomorphism between fundamental groups at least when the special fiber is supersingular.