Arithmetic Analogs of the Standard Conjectures

摘要

The paper proposes that all standard conjectures have analogs for the 'arithmeticChow groups' of arbitrary arithmetic varieties. After recalling arithmetic intersection formalism, it conjectures that the arithmetic intersection pairing is nondegenerate. This fits nicely with conjectures about heights and regulators. It proposes also a hard Lefschetz and a Hodge index type statement. These conjectures are true for arithmetic surfaces. It then gives a definition of arithmetic correspondences based on the notion of 'regular kernels'. It is interesting to notice that the arithmetic correspondences from X to Y do not consist only of arithmetic cycles on X x Y. However, the notion of correspondences is still too restrictive, and the authors have no analog of the assertion that the different operators coming from Lefschetz decomposition (e.g. the star operator) are induced by correspondences. The reason is that, in the absence of a base point, the product X x Y has to be taken over a base of dimension one, and their correspondences are relative to the base. The attempt to combine Kahler geometry with intersection theory will help in gaining some insight on these questions.