Mordell-Weil Lattices of Elliptic Curves and Abelian Varieties

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(1) Integral Points and Mordell-Weil Lattices.As an application of Mordell-Weil Lattices, we have developed a method to study integral points in the function field case. In some favorable situation, this method gives a very efficient way for a complete determination of all the integral points of an elliptic curve. [S1](2) K3 Surfaces and Sphere PackingsWe hare obtained lattice sphere packings in higher dimensional case (especially dimension 16, 17, 18) of fairly large packing density, by means of the Mordell-Weil Lattices of certain elliptic K3 surfaces. [S3](3) Invariant theory of plane quartics vs Mordell-Weil LatticesWe hare established a close relationship of the classical invariant theory of plane quartics (moduli of genus three curves) and the invariant theory of the Weyl group of type E_7 (a finite group). [S4](4) Some codes arising from the elliptic modular surfacesFor any N. we have constructed a linear code over the residue ring mod N which is associated with the elliptic modular surfaces of level N. If N is a prime number, this linear code over a field of N elements has a remarkable property that every nonzero code-word has a constant Bernoulli norm. The construction is based on the height formula of Mordell-Weil Lattices, [S2](5) Tate-Shafarevich group of elliptic curvesAoki has proven that the 3-part of Tate-Shafarevich group can be arbitrarily large. [A2](6) Hodge conjecture of abelian varietiesThe Hodge cycles on the Jacobian variety of a Fermat curve are studied from combinatorial viewpoint. By this, the Hodge conjecture is verified for wider class of abelian varieties of Fermat type. [A1], [A3]