Weil numbers and generating large field extensions.

The first half of this thesis involves the study of Weil-numbers and their properties. Using characters satisfying the Serre condition we construct a large class of Weil-numbers. We also study two tori associated to a Weil-number $\pi$: $L(\pi)$ and $S(\pi)$. We show that if an abelian variety corresponds, using the theorem of Tate and Honda, to a Weil-number $\pi$ such that $L(\pi) \not= S(\pi)$ then the abelian variety has exceptional Tate cycles. The second half of this thesis is a partial response to Hilbert's 12th Problem: generating abelian extensions of number fields. Using the theory of Shimura varieties we generate large abelian extensions of CM-fields. We also explain how these fields are generated by special values of modular functions on Shimura varieties.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/103874/1/9409835.pdfDescription of 9409835.pdf : Restricted to UM users only