### 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) |
---|---|

Pages (from-to) | 111-150 |

Number of pages | 40 |

Journal | Algebra and Number Theory |

Volume | 4 |

Issue number | 2 |

State | Published - 2010 |

Externally published | Yes |

### Fingerprint

### 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

### Cite this

*Algebra and Number Theory*,

*4*(2), 111-150.

**Canonical extensions of néron models of jacobians.** / Cais, Bryden R.

Research output: Contribution to journal › Article

*Algebra and Number Theory*, vol. 4, no. 2, pp. 111-150.

}

TY - JOUR

T1 - Canonical extensions of néron models of jacobians

AU - Cais, Bryden R

PY - 2010

Y1 - 2010

N2 - Let A be the néron model of an abelian variety Aκ 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 Aκ = Jκ is the Jacobian of a smooth, proper and geometrically connected curve Xκ over K. Assuming that Xκ 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,0x/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 Pic0x/r. As an application of our result, we prove a comparison isomorphism between two canonical integral structures on the de Rham cohomology of Xκ.

AB - Let A be the néron model of an abelian variety Aκ 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 Aκ = Jκ is the Jacobian of a smooth, proper and geometrically connected curve Xκ over K. Assuming that Xκ 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,0x/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 Pic0x/r. As an application of our result, we prove a comparison isomorphism between two canonical integral structures on the de Rham cohomology of Xκ.

KW - Abelian variety

KW - Canonical extensions

KW - De Rham cohomology

KW - Grothendieck duality

KW - Grothendieck's pairing

KW - Group schemes

KW - Integral structure

KW - Lacobians

KW - Neéon models

KW - Relative picard functor

KW - Rigidifled extensions

UR - http://www.scopus.com/inward/record.url?scp=77953977319&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953977319&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:77953977319

VL - 4

SP - 111

EP - 150

JO - Algebra and Number Theory

JF - Algebra and Number Theory

SN - 1937-0652

IS - 2

ER -