Arithmetic and Geometry Abelian Varieties

研究者和他的同事研究了低维阿贝尔变上的Hodge和Tate类，并特别提到了用线性代数从除数类和Weil类中得到所有这些类的可能性。他们研究了数域上无主极化的阿贝尔变异的同源类，并特别提到了这些同源类的显式构造，这些同源类只携带其度可被给定正整数整除的极化。研究了超椭圆雅可比矩阵和三角雅可比矩阵的自同态环。该项目涉及与所谓的阿贝尔积分相关的各种对称性。这些积分及其隐藏的对称性可能有助于构建更有效的代码，以及写出重要方程的显式解。这个项目的目的是控制某些隐藏的对称性，当尺寸很小的时候，在任意尺寸下，提供明确的结构，只有一些额外的对称性。

Abstract--ZarhinThe investigator and his colleagues study Hodge and Tate classes on abelian varieties of low dimension with special reference to the possibility of obtaining all these classes from divisor classes and Weil classes by means of linear algebra. They study isogeny classes of abelian varieties over number fields without principal polarizations with special reference to explicit constructions of such isogeny classes that carry only polarizations whose degree is divisible by a given positive integer. The investigator studies the endomorphism rings of hyperelliptic and trigonal jacobians.The project deals with various symmetries associated with so-called abelian integrals. These integrals and their hidden symmetries may be of help in constructing more efficient codes, as well as in writing down explicit solutions to important equations. The aim of this project is to control certain hidden symmetries when the dimension is small, and in arbitrary dimensions, to provide explicit constructions with just a few additional symmetries.