Arithmetics on Jacobian Varieties

1. 定义在有限域上的超椭圆曲线的雅可比变的分类：我们尝试(1)对定义在q阶过有限域上的g属超椭圆曲线进行分类，然后(2)对特定情况下由(1)分类的曲线的雅可比变进行分类。在q = 3,5的情况下，对所有属2的超椭圆曲线进行分类，并通过计算每条曲线的同余zeta函数得到同系类。然后确定了每个雅可比矩阵的自态环。具有大阶有理点的阿贝尔变的构造：(1)超椭圆曲线雅可比变上特定点的阶数可由曲线上相应函数的连续分数展开周期给出。(2)在有理域上得到了一类2型超椭圆曲线，其雅可比变分有23.3阶点。3 .关于阿贝尔变量的n分域的伽罗瓦群的计算：对于gunus 2、2和3分方程的超椭圆曲线的雅可比变量，通过计算过程确定。多有理点代数曲线的构造：利用雅可比矩阵变换的n除法方程，得到了函数域上多有理点代数曲线的级数。

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.