## Doctoral Dissertations

Dissertations that have an embargo placed on them will not be available to anyone until the embargo expires.

N/A

#### Document Type

Open Access Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

2014

February

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.

#### DOI

https://doi.org/10.7275/pjwq-6705

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