Abelian Surfaces with Complex Multiplication Admitting Nonprincipal Polarizations

Abstract

The theory of complex multiplication of abelian varieties is a useful field of study with applications ranging from the explicit construction of abelian extensions CM-fields to the explicit description of L-functions of abelian varieties in ways which are much easier to carry out than the more general case. In the literature the most commonly studied abelian surfaces are those with a principal polarization. In the present thesis we extend this analysis to describe abelian surfaces with complex multiplication which carry a nonprincipal polarization. We provide a complete characterization of which types of polarizations are possible on abelian surfaces which have complex multiplication by a given quartic CM-field K as well as how to construct them when they do exist. We also derive several necessary conditions for such abelian surfaces to exist as well as provide an existence theorem in limited circumstances.doctoral, Ph.D., Mathematics & Statistical Sci -- University of Idaho - College of Graduate Studies, 2023-0