Arakelov Chow groups of abelian schemes, arithmetic Fourier transform, and analogues of the standard conjectures of Lefschetz type
We discuss absolute and relative analogues of Grothendieck's standard conjectures of Lefschetz type for the arithmetic Chow groups introduced by Gillet and Soulé. We decompose the Arakelov Chow groups of an abelian scheme into eigenspaces with respect to \(n\)-multiplication and investigate their natural ring of correspondences. The main technical tool to obtain these results is an arithmetic analogue of the algebraic theory of the Fourier transform. We discuss filtrations on the arithmetic Chow groups and describe how the absolute hard Lefschetz conjecture of Gillet and Soulé can be derived from a relative version and a conjecture of Beilinson about the images of regulator maps in Deligne cohomology. Finally, we compute the cubical height of divisors on abelian schemes.
Grothendieck's standard conjectures of Lefschetz type
filtrations on the arithmetic Chow groups
scientific article; zbMATH DE number 701004
scientific article