Mordell-Weil Lattices of Elliptic Curves and Abelian Varieties

(1)整点与Mordell-Weil格作为Mordell-Weil格的一个应用，我们发展了一种研究函数域中整点的方法。在某些有利的情况下，该方法为求椭圆曲线的所有整点提供了一种非常有效的方法。（2）K_3曲面与球填充利用某些椭圆K_3曲面的Mordell-Weil格，我们得到了高维（特别是16，17，18维）情况下具有较大填充密度的格球填充。（3）平面四次线的不变理论与Mordell-Weil格我们建立了经典的平面四次线的不变理论（亏格为三条曲线的模）与E_7型Weyl群（有限群）的不变理论之间的密切关系。[S4]（4）由椭圆模曲面产生的一些码。我们构造了模N剩余环上的一个线性码，它与N阶椭圆模曲面相联系。如果N是一个素数，这个在N个元素的域上的线性码有一个显著的性质，即每个非零码字都有一个常数伯努利范数。[S2]（5）椭圆曲线的Tate-Shafarevich群Aoki证明了Tate-Shafarevich群的3-部分可以是任意大的。[A2]（6）交换簇的Hodge猜想从组合的观点研究了Fermat曲线的Jacobi簇上的Hodge圈。由此，Hodge猜想在更广的一类Fermat型交换簇上得到了验证。[A1]、[A3]

(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]