### Abstract

Let A be the néron model of an abelian variety A_{κ} 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 _{κ} 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_{κ} = J_{κ} is the Jacobian of a smooth, proper and geometrically connected curve X_{κ} over K. Assuming that X_{κ} 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 ^{b,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_{κ}.

Original language | English (US) |
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Pages (from-to) | 111-150 |

Number of pages | 40 |

Journal | Algebra and Number Theory |

Volume | 4 |

Issue number | 2 |

DOIs | |

State | Published - 2010 |

Externally published | Yes |

### Keywords

- Abelian variety
- Canonical extensions
- De Rham cohomology
- Grothendieck duality
- Grothendieck's pairing
- Group schemes
- Integral structure
- Lacobians
- Neéon models
- Relative picard functor
- Rigidifled extensions

### ASJC Scopus subject areas

- Algebra and Number Theory