Mordell-Weil Lattices and Cycles on Algebraic Surfaces

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The following results have been accomplished on Algebraic cycles on K3 surfaces which forms the main theme of this research proposal :(i) We have determined, for an example of a K3 surface attaining the maximal Mordell-Weil rank 18, the structure of its Mordell-Weil lattice, explicit generators of rational points, together with the splitting field.[2](ii) More generally, we have clarified the structure of the transcendental lattice, Neron-Severi lattice and the Mordell-Weil lattice, of the singular K3 surfaces which arise via the cyclic base change of degree up to six from Inose's elliptic K3 surface. Ref.[3] and in preparation, and (3).(iii) Classical Kummer surfaces attached to genus two Jacobian varieties are studied from the viewpoint of Mordell-Weil lattices, which establishes the close relationship of automorphisms of a genus two curve with the singular fibres and generators of the Mordell-Weil lattice in question. Ref.[4].(iv) In a joint work with M.Kuwata, we have determined the elliptic parameters and the defining equations of elliptic fibrations on the Kummer surface of the product of two non-isogenous elliptic crves., which has been classified by Oguiso by geometric method. Ref.[6].[The rest is omitted in this English summary because of the space limitation.]