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Document Type

Open Access Dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Mathematics

Year Degree Awarded

Spring 2014

First Advisor

Siman Wong

Abstract

Let $F$ be a number field and let $A$ be an abelian algebraic group defined over $F$. For a prime $\ell$ and a point $\alpha \in A(F)$, we obtain the tower of extensions $F([\ell^n]^{-1}(\alpha))$ by adjoining to $F$ the coordinates of all the preimages of $\alpha$ under multiplication by $[\ell^n]$. This tower contains the coordinates of all of the $\ell$-power torsion points of $A$ along with a Kummer-type extension. The Galois groups of these extensions encode information about the density of primes $\cP$ in the ring of integers of $F$ for which the order of $\alpha$ (mod $\cP$) is not divisible by $\ell$. In this thesis, we determine these Galois groups and explicitly compute the associated density for the cases where $A$ is (1) a reducible elliptic curve; (2) a product of elliptic curves with complex multiplication; (3) an abelian surface with real multiplication.

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