Arithmetics on Jacobian Varieties

1.定义在有限域上的超椭圆曲线的Jacobi簇的分类：我们试图（1）对定义在q阶有限域上的亏格为g的超椭圆曲线进行分类，然后（2）在特殊情况下对由（1）分类的曲线的Jacobi簇进行分类。当q = 3，5时，对亏格为2的超椭圆曲线进行了分类，并通过计算每条曲线的同余zeta函数得到了等距类.然后确定了每个Jacobian的自同态环.具有高阶有理点的交换簇的构造：（1）双椭圆曲线的Jacobi簇上特定点的阶可由相应函数在曲线上的连分式展开的周期给出。(2)我们发现了亏格2在有理数域上的一系列超椭圆曲线，其雅可比簇的阶数为23.3。交换簇的n-可除域的伽罗瓦群的计算：对于Gunus 2的超椭圆曲线的雅可比簇，通过计算处理，确定了2-可除和3-可除方程。多有理点代数曲线的构造：利用Jacobian簇的n-除法方程，得到了多有理点函数域上的代数曲线序列。

1. Classification of Jacobian varieties of hyperelliptic curves defined over finite fields : We tried to (1) classify hyperelliptic cureves of genus g defined overfinite field of order q then (2) classify Jacobian varieties of curves classifyed by (1) in specific cases. In case q = 3,5 all hyperelliptic curves of genus 2 are classified and get isogeny classes by computing congruence zeta fanction of each curve. Then we determied the endmorphism rings of each Jacobians.2. Construction of abelian varieties of with a rational point of large order : (1) Orders of specific points on Jacobian varieties of hperelliptic curves can be given by the periods of continued fractional expansion of corrsponding functions on the curves. (2) We found a series of hyperelliptic curves of geneus 2 over the rationalfield, whose Jacobian varieties have a point of order 23.3. Computation of Galois groups of the n-division fields of abelian varieties : For the Jacobian varieties of hyperelliptic curves of gunus 2,2 and 3-division equations are determied by computational-procession.4. Construction of algebraic curves with many ration points : We got some series of algebraic curves over funcion fields with many rational points, by using the n-division equation of Jacobian varieties.