ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11, 14 Logachev, Dmitry, Moscow, Russia Mazur - Bloch method for Siegel threefolds Let X be a Siegel threefold and V ∈ CH^{2}(X)(K)_{0} a Heegner cycle homologically equivalent to 0. Let cl′(V) ∈ H^{1}(K, H^{3}(X, ℤ_{l}(2))) be its Abel-Jacobi image (a high-dimensional analog of a Heegner point). By analogy with one-dimensional case we can expect that (*) In many cases cl′(V) is not of torsion. We give an evidence that Mazur-Bloch method can be used to prove (*). First steps of application of this method for the present situation are described. The main step of Mazur-Bloch method is finding of a group Φ(H^{3}(X)), and of the image of V in Φ. It is shown how to find this image under some simplifying assumptions. An important step that has to be made is finding of a desingularization of a model (constructed by de Jong) of X over ℤ_{p}.