Canonical extensions of néron models of jacobians

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)111-150
Number of pages40
JournalAlgebra and Number Theory
Volume4
Issue number2
StatePublished - 2010
Externally publishedYes

Fingerprint

Canonical Extension
Functor
Fiber
De Rham Cohomology
Valuation Ring
Group Scheme
Abelian Variety
Line Bundle
Natural Extension
Isomorphism
Model
Partial
Closed
Curve
Zero
Theorem

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

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

In: Algebra and Number Theory, Vol. 4, No. 2, 2010, p. 111-150.

Research output: Contribution to journalArticle

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