Experimental Method of Mathematics

我们利用计算机代数软件，在算术、代数几何、自同构函数、黎曼曲面和密码系统等领域进行了大量的工作，并得到了许多有趣的结果，其中一些结果如下：代数数论：(1)类数和数域基本单位的计算。(2)实二次域的若干残余类群与虚二次域的类群之间的关系。(3)实数二次域的Kronecker极限公式的数值计算。(4)有理数域上二面体扩展的单位群的群结构。有理域上椭圆曲线的算法：(1)正则参数和正则幂级数解的存在性；(2)自同构函数对椭圆曲线的均匀化。(3)有理数环上正则幂级数和全局ζ函数的算法计算。(4)椭圆曲线上正则算术结构的存在性。复数函数三维图形表示的应用：(1)用complex Plot命令表示黎曼曲面。(2)研究由幂级数给出的复函数的周期性质。(3)椭圆曲线正则幂序列的自同构性质。

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.