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2010 Canonical extensions of Néron models of Jacobians
Bryden Cais
Algebra Number Theory 4(2): 111-150 (2010). DOI: 10.2140/ant.2010.4.111


Let A be the Néron model of an abelian variety AK 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 AK 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 AK=JK is the Jacobian of a smooth, proper and geometrically connected curve XK over K. Assuming that XK 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 PicXR,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 JK with the functor PicXR0. As an application of our result, we prove a comparison isomorphism between two canonical integral structures on the de Rham cohomology of XK.


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Bryden Cais. "Canonical extensions of Néron models of Jacobians." Algebra Number Theory 4 (2) 111 - 150, 2010.


Received: 28 October 2008; Revised: 18 July 2009; Accepted: 15 August 2009; Published: 2010
First available in Project Euclid: 20 December 2017

zbMATH: 1193.14058
MathSciNet: MR2592016
Digital Object Identifier: 10.2140/ant.2010.4.111

Primary: 14L15
Secondary: 11G20 , 14F30 , 14F40 , 14H30 , 14K30

Keywords: abelian variety , canonical extensions , de Rham cohomology , Grothendieck duality , Grothendieck's pairing , group schemes , integral structure , Jacobians , Néron models , relative Picard functor , rigidified extensions

Rights: Copyright © 2010 Mathematical Sciences Publishers


Vol.4 • No. 2 • 2010
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