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 $R$. By work of Mazur and 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 $A$. Here 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 ${Pic}_{X∕R}^{♮,0}$ 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 Néron model $J$ of $J_K$ with the functor ${Pic}_{X∕R}^0$. As an application of our result, we prove a comparison isomorphism between two canonical integral structures on the de Rham cohomology of $X_K$.