Canonical extensions of Néron models of Jacobians

Abstract

Let A be the Néron model of an abelian variety A_K over the fraction field K of a discrete valuation ring R. Due to work of Mazur-Messing, there is a functorial way to prolong the universal extension of A_K by a vector group to a smooth and separated group scheme over R, called the canonical extension of A. In this paper, we study the canonical extension when A_K=J_K is the Jacobian of a smooth proper and geometrically connected curve X_K over K. Assuming that X_K 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 Pic^{\natural,0}_{X/R} 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 Néron model J of J_K with the functor Pic^0_{X/R}. As an application of our result, we prove a comparison isomorphism between two canonical integral structures on the de Rham cohomology of X_K.

To appear in Algebra & Number Theory