Algebra & Number Theory

Néron's pairing and relative algebraic equivalence

Cédric Pépin

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Let R be a complete discrete valuation ring with algebraically closed residue field k and fraction field K. Let XK be a proper smooth and geometrically connected scheme over K. Néron defined a canonical pairing on XK between 0-cycles of degree zero and divisors which are algebraically equivalent to zero. When XK is an abelian variety, and if one restricts to those 0-cycles supported on K-rational points, Néron gave an expression of his pairing involving intersection multiplicities on the Néron model A of AK over R. When XK is a curve, Gross and Hriljac gave independently an analogous description of Néron’s pairing, but for arbitrary 0-cycles of degree zero, by means of intersection theory on a proper flat regular R-model X of XK.

We show that these intersection computations are valid for an arbitrary scheme XK as above and arbitrary 0-cycles of degree zero, by using a proper flat normal and semifactorial model X of XK over R. When XK=AK is an abelian variety, and X=A¯ is a semifactorial compactification of its Néron model A, these computations can be used to study the relative algebraic equivalence on A¯R. We then obtain an interpretation of Grothendieck’s duality for the Néron model A, in terms of the Picard functor of A¯ over R. Finally, we give an explicit description of Grothendieck’s duality pairing when AK is the Jacobian of a curve of index one.

Article information

Algebra Number Theory, Volume 6, Number 7 (2012), 1315-1348.

Received: 19 February 2011
Revised: 21 December 2011
Accepted: 18 January 2012
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 14K30: Picard schemes, higher Jacobians [See also 14H40, 32G20]
Secondary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30] 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx] 11G10: Abelian varieties of dimension > 1 [See also 14Kxx]

Néron's symbol Picard functor Néron models duality Grothendieck's pairing


Pépin, Cédric. Néron's pairing and relative algebraic equivalence. Algebra Number Theory 6 (2012), no. 7, 1315--1348. doi:10.2140/ant.2012.6.1315.

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