Vol. 4, No. 2, 2010

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Canonical extensions of Néron models of Jacobians

Bryden Cais

Vol. 4 (2010), No. 2, 111–150

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.

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
Received: 28 October 2008
Revised: 18 July 2009
Accepted: 15 August 2009
Published: 26 January 2010
Bryden Cais
Department of Mathematics and Statistics
McGill University
805 Sherbrooke St. West
Montréal, QC H3A 2K6