Cohomological arithmetic Chow rings
Abstract
We develop a theory of abstract arithmetic Chow rings where the role of the fibers at infinity is played by a complex of abelian groups that computes a suitable cohomology theory. This theory allows the construction of many variants of the arithmetic Chow groups with different properties. As particular cases of this formalism we recover the original arithmetic intersection theory of Gillet and Soulé for projective varieties, we introduce a theory of arithmetic Chow groups which are covariant with respect to arbitrary proper morphisms, and we develop a theory of arithmetic Chow rings using a complex of differential forms with loglog singularities along a fixed normal crossings divisor. This last theory is suitable for the study of automorphic line bundles. In particular, we generalize the classical Faltings height with respect to a logarithmically singular hermitian line bundle to higher dimensional cycles. As an application we compute the Faltings height of Hecke correspondences on a product of modular curves.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 April 2004
 arXiv:
 arXiv:math/0404122
 Bibcode:
 2004math......4122B
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Algebraic Geometry;
 14G40 14G35 14C17 14C30 11G18
 EPrint:
 Minor errors and typos corrected