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Author ORCID Identifier
N/A
AccessType
Open Access Dissertation
Document Type
dissertation
Degree Name
Doctor of Philosophy (PhD)
Degree Program
Mathematics
Year Degree Awarded
2014
Month Degree Awarded
February
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.
DOI
https://doi.org/10.7275/pjwq-6705
Recommended Citation
Aiello, Domenico, "GALOIS THEORY OF ITERATED MORPHISMS ON REDUCIBLE ELLIPTIC CURVES AND ABELIAN SURFACES WITH REAL MULTIPLICATION" (2014). Doctoral Dissertations. 46.
https://doi.org/10.7275/pjwq-6705
https://scholarworks.umass.edu/dissertations_2/46