Canonical extensions of Néron models of Jacobians

Abstract

Let A be the Neron model of an abelian variety AK over the fraction field K of a discrete valuation ring R. By work of Mazur and Messing, there is a functorial way to prolong the universal extension of AK by a vector group to a smooth and separated group scheme over R, called the canonical extension of A. Here we study the canonical extension when AK=JK is the Jacobian of a smooth, proper and geometrically connected curve XK over K. Assuming that XK admits a proper flat regular model X over R that has generically smooth closed fiber, our main result identifies the identity component of the canonical extension with a certain functor PicX∕R♮,0 classifying line bundles on X that have partial degree zero on all components of geometric fibers and are equipped with a regular connection. This result is a natural extension of a theorem of Raynaud, which identifies the identity component of the Neron model J of JK with the functor PicX∕R0. As an application of our result, we prove a comparison isomorphism between two canonical integral structures on the de Rham cohomology of XK.