Arithmetic Chow groups

General Introduction
Defined by GilletSoulé.
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Online References
See Burgos Gil, Kramer, Kuehn and references therein.
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Paper References
Soulé, Abramovich, Burnol and Kramer: Lectures on Arakelov Geometry. Especially chapter II and III.
Must take into account Bloch, Gillet, Soulé: Nonarch. Arakelov theory, in JAG 4 no 3. They define arithmetic Chow groups for nonarch places. MR1325788
Gillet and Soulé: Arithmetic analogues of the standard conjectures (in Motives vol)
Goncharov in Ktheory 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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Definition
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
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 pullback and pushforward 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 selfadjoint 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 selfintersection of the dualizing sheaf. Using the GrossZagier formula and a calculation due to U. Kuehn we compute these invariants in terms of special values of L series.
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