Formal completions of Néron models for algebraic tori

Oleg Demchenko; Alexander Gurevich; Xavier Xarles

doi:10.1112/plms/pdp039

Formal completions of Néron models for algebraic tori

Proceedings of the London Mathematical Society ◽

10.1112/plms/pdp039 ◽

2009 ◽

Vol 100 (3) ◽

pp. 607-638 ◽

Cited By ~ 2

Author(s):

Oleg Demchenko ◽

Alexander Gurevich ◽

Xavier Xarles

Keyword(s):

Algebraic Tori ◽

Neron Models ◽

Néron Models

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Jumps and Motivic Invariants of Semiabelian Jacobians

International Mathematics Research Notices ◽

10.1093/imrn/rnx303 ◽

2018 ◽

Vol 2019 (20) ◽

pp. 6437-6479

Author(s):

Otto Overkamp

Keyword(s):

Exact Sequence ◽

A Priori ◽

Smooth Curve ◽

Finite Sequence ◽

Base Change ◽

Algebraic Tori ◽

Neron Models ◽

Picard Functor ◽

Separable Extensions ◽

Néron Models

Abstract We investigate Néron models of Jacobians of singular curves over strictly Henselian discretely valued fields and their behavior under tame base change. For a semiabelian variety, this behavior is governed by a finite sequence of (a priori) real numbers between 0 and 1, called jumps. The jumps are conjectured to be rational, which is known in some cases. The purpose of this paper is to prove this conjecture in the case where the semiabelian variety is the Jacobian of a geometrically integral curve with a push-out singularity. Along the way, we prove the conjecture for algebraic tori which are induced along finite separable extensions and generalize Raynaud’s description of the identity component of the Néron model of the Jacobian of a smooth curve (in terms of the Picard functor of a proper, flat, and regular model) to our situation. The main technical result of this paper is that the exact sequence that decomposes the Jacobian of one of our singular curves into its toric and Abelian parts extends to an exact sequence of Néron models. Previously, only split semiabelian varieties were known to have this property.

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Crystalline subrepresentations and Neron models

Mathematical Research Letters ◽

10.4310/mrl.2000.v7.n5.a6 ◽

2000 ◽

Vol 7 (5) ◽

pp. 605-614

Author(s):

Minhyong Kim ◽

Susan H. Marshall

Keyword(s):

Neron Models ◽

Néron Models

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Néron models

European Journal of Mathematics ◽

10.1007/s40879-017-0134-8 ◽

2017 ◽

Vol 3 (2) ◽

pp. 171-198

Author(s):

Dino Lorenzini

Keyword(s):

Neron Models ◽

Néron Models

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Néron models and limits of Abel–Jacobi mappings

Compositio Mathematica ◽

10.1112/s0010437x09004400 ◽

2010 ◽

Vol 146 (2) ◽

pp. 288-366 ◽

Cited By ~ 6

Author(s):

Mark Green ◽

Phillip Griffiths ◽

Matt Kerr

Keyword(s):

Normal Function ◽

Motivic Cohomology ◽

Normal Functions ◽

Geometric Setting ◽

Neron Models ◽

A Finite Group ◽

Intermediate Jacobian ◽

Néron Models ◽

Singular Fibre ◽

Finite Singularity

AbstractWe show that the limit of a one-parameter admissible normal function with no singularities lies in a non-classical sub-object of the limiting intermediate Jacobian. Using this, we construct a Hausdorff slit analytic space, with complex Lie group fibres, which ‘graphs’ such normal functions. For singular normal functions, an extension of the sub-object by a finite group leads to the Néron models. When the normal function comes from geometry, that is, a family of algebraic cycles on a semistably degenerating family of varieties, its limit may be interpreted via the Abel–Jacobi map on motivic cohomology of the singular fibre, hence via regulators onK-groups of its substrata. Two examples are worked out in detail, for families of 1-cycles on CY and abelian 3-folds, where this produces interesting arithmetic constraints on such limits. We also show how to compute the finite ‘singularity group’ in the geometric setting.

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