Arithmetic and Geometry Abelian Varieties

Pennsylvania State Univ University Park, University Park PA

Investigators

Abstract

Abstract--Zarhin The 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.