Crystalline representations and Neron models

Abstract

We define and study the maximal crystalline subrepresentation functor, Crys(-), defined on p-adic Galois representations of the absolute Galois group of a finite extension K of Q(p) . In particular, we define and study the derived functors, Rⁱ Crys(-), of Crys(-). We then apply these functors to the study of Neron models of abelian varieties defined over K. We extend a formula of Grothendieck expressing the component group of a Neron model in terms of Galois cohomology. The extended formula is only valid for abelian varieties with semistable reduction defined over an unramified base. We explore the failure of the formula in the non-semistable case through the example furnished by Jacobians of Fermat curves.