Shafarevich Correspondence and Mordell-Weil Lattices

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The idea of Shafarevich correspondence makes it possible to relate the elliptic curves having given discriminant with the integral points of the partner elliptic curve. Combined with the theory of Mordell-Weil lattices, this idea enables the principal investigator to obtain the following results :(1)Determination of K3 surface with a maximal singular fibre(2)Application of Davenport-Stothers triples to elliptic surfaces(3)Abc-theorem and the number of singular fibres of an elliptic surface over P^1On the other hand, the investigator Aoki have studied the arithmetic of elliptic curves and abelian varieties over a number field, as well as the Hodge conjecture. More specifi cally, he has treated the following subjects :(4)Tate-Shafarevich groups of semistable elliptic curves with a rational 3-torsion(5)Hodge conjecture for the jacobian varieties of generalized Catalan curves,(6)On the generalized Gross-Tate conjecture for elementary abelian 2-extensions