#### Vol. 4, No. 2, 2010

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editors' Interests Submission Guidelines Submission Form Editorial Login Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
Canonical extensions of Néron models of Jacobians

### Bryden Cais

Vol. 4 (2010), No. 2, 111–150
##### 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 ${X}_{K}$.

##### Keywords
canonical extensions, Néron models, Jacobians, relative Picard functor, group schemes, Grothendieck's pairing, Grothendieck duality, integral structure, de Rham cohomology, abelian variety, rigidified extensions
##### Mathematical Subject Classification 2000
Primary: 14L15
Secondary: 14F30, 14F40, 11G20, 14K30, 14H30