Experimental Method of Mathematics

我们利用计算机代数软件，主要在算术、代数几何、自守函数、黎曼曲面和密码体制等方面进行了工作，得到了许多有趣的结果，其中包括：1.代数数论：（1）数域的类数和基本单位的计算。(2)真实的二次域的某些剩余类群与虚二次域的类群之间的关系。(3)真实的二次域Kronecker极限公式的数值计算(4)有理数域上二面体扩张单位群的群结构.定义在有理数域上的椭圆曲线的算术：（1）标准参数和标准幂级数解的存在性，（2）用自守函数对椭圆曲线进行统一。(3)有理整数环上的标准幂级数和整体zeta函数的数学计算。(4)椭圆曲线上正则算术结构的存在性.复变函数三维图形表示的应用：（1）用Complex Plot命令表示Riemann曲面。(2)研究由幂级数给出的复变函数的周期性。(3)椭圆曲线标准幂级数的自守性质。

We worked, by usisng softwares for the computer algebra, manily in the fields of arithmetics, algebraic geometry, automorphic functions, Riemann surfaces and cryptosystems and gots many interesting results, some of them are as follows :1. Algebraic Number Thoery : (1) Computation of classnumbers and fundamental units of number fields. (2) Relations between certain residule class groups of real quadratic fields and class groups of imaginary quadratic fields. (3) Numerical computations of Kronecker's limit-formula of real quadratic fields. (4) Group structures of the unit groups of Dihedral extensions over the rational number field.2. Arithmetics in elliptic curves defined over the rational field : (1) Existence of the canonical parameters and canonical power series solutions, (2) Uniformizaton of elliptic curves by automorphic functions. (3) Algorithmic Computation of canonical power series and the global zeta-function over the ring of rational integers. (4) Existence of canonical arithmetic-structures in elliptic curves.3. Application of 3D-graphical representation of complex functions : (1) Represatations of Riemann surfaces by Complex Plot commands. (2) Investigating periodical properties of complex functions given by power series. (3) Automorphic properties of canonical power seires of elliptic curves.