Holmstrom Arithmetic Chow groups

Arithmetic Chow groups

Defined by Gillet-Soule.

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Arithmetic Chow groups

See Burgos Gil, Kramer, Kuehn and references therein.

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Arithmetic Chow groups

Roberts: Multiplicities and Chern classes in local algebra. A review. This looks like a nice treatment of intersection theory for very general schemes, possibly interesting in number theory. See also Serre: Local algebra

Burgos: Arithmetic Chow rings and Deligne-Beilinson cohomology, 1997.

Gillet and Soulé: Arithmetic intersection theory (1991), IHES.

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Arithmetic Chow groups

MathSciNet

Google Scholar

Google

arXiv: Experimental full text search

arXiv: Abstract search

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Arithmetic Chow groups

AAG (Arithmetic algebraic geometry)

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Arithmetic Chow groups

Arithmetic

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Arithmetic Chow groups

Soulé, Abramovich, Burnol and Kramer: Lectures on Arakelov Geometry. Especially chapter II and III.

Must take into account Bloch, Gillet, Soule: Nonarch. Arakelov theory, in JAG 4 no 3. They define arithmetic Chow groups for nonarch places. http://www.ams.org/mathscinet-getitem?mr=1325788

Gillet and Soulé: Arithmetic analogues of the standard conjectures (in Motives vol)

Goncharov in K-theory handbook, p. 312.

See also Maillot: G´eom´etrie d’Arakelov des vari´et´es toriques et fibr´es en droites int´egrables, for the notion of generalized arithmetic Chow groups, appropriate for Chern classes of bundles with singular metric.

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Arithmetic Chow groups

arXiv:0911.0546 Correspondences in Arakelov geometry. Applications to Hecke operators on modular curves from arXiv Front: math.AG by Ricardo Menares In the context of arithmetic surfaces, J.-B. Bost defined a generalized Arithmetic Chow Group (ACG) using the Sobolev space L^2_1. We study the behavior of these groups under pull-back and push-forward and we prove a projection formula. We use these results to define an action of the Hecke operators on the ACG of modular curves and show that they are self-adjoint with respect to the arithmetic intersection product. The decomposition of the ACG in eigencomponents which follows allows us to define new numerical invariants, which are refined versions of the self-intersection of the dualizing sheaf. Using the Gross-Zagier formula and a calculation due to U. Kuehn we compute these invariants in terms of special values of L series.

nLab page on Arithmetic Chow groups