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In this thesis, we investigate modularity of elliptic curves over a general totally real number field, establishing a finiteness result for the set non-modular j-invariants. By analyzing quadratic points on some modular curves, we show that all elliptic curves over certain real quadratic fields are modular. / Mathematics

published_or_final_version / Mathematics / Master / Master of Philosophy

This study investigated domains within the tasks of ordination, cardination and natural number. In addition, it examined the sequencing of the development of established domains between ordination, cardination and natural number. One hundred and forty-eight children were individually given a test designed to ascertain the presence of arithmetic related skills. The task of cardination was designed to detect the ability to associate numbers with sets containing a group of elements ranging from one to five. The task of ordination was designed to detect the ability to associate a number with a relative position within a group of ordered objects. The task of natural number assessed children's ability to add numbers with a sum equal to five or less. Latent structure analysis was used to analyze the results. Four different models were used in order to establish domains within each task. The four models that were used tested independence, equiprobability, ordered relations and asymmetrical equivalence. The results showed two domains for cardination. One item in a set constituted one domain while three and five items in a set constituted a second domain. The domain for ordination included relative positions one, three and five. In regards to natural number, the results showed a permeable domain. There was some indication of ordering but it was not strong enough to yield separate domains. The same models were then used to compare across the different tasks. The results showed that the easiest cardination domain developed before ordination and natural number. The results also showed that the ordination task was equivalent to one of the natural number tasks. All of the other comparisons between cardination, ordination and natural number yielded asymmetrically equivalent relations. That is, there was an ordering but the ordering was not strong enough to establish separate domains.

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Two of the more well known congruences for binomial coefficients modulo p, due to Gauss and Jacobi, are related to the representation of an odd prime (or an integer multiple of the odd prime) p as a sum of two squares (or an integer linear combination of two squares). These two congruences, along with many others, have been extended to analogues modulo p^2 and are well documented. More recently, J. Cosgrave and K. Dilcher have extended the congruences of Gauss and Jacobi to analogues modulo p^3. In this thesis we discuss their methods as well as the potential of applying them to similar congruences found in the literature.

Using a classical result of Thue, we give an upper bound for the number of solutions to a family of quartic Thue equations. We also give an upper bound upon the number of solutions to a family of quartic Thue inequalities. Using the Thue-Siegel principle and the theory of linear forms in logarithms, an upper bound is given for general quartic Thue equations. As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation aX⁴ - bY² = 1, for fixed positive integers a and b, possesses at most two solutions in positive integers X and Y. Since there are infinitely many pairs (a, b) for which two such solutions exist, this result is sharp. It is also effectively proved that for fixed positive integers a and b, there are at most two positive integer solutions to the quartic Diophantine equation aX⁴ - bY² = 2. We will also study cubic and quartic Thue equations by combining some classical methods from Diophantine analysis with modern geometric ideas.