Thesis (M.S.)--Youngstown State University, 2009. / Includes index. Also available via the World Wide Web in PDF format.

Thesis--Copenhagen. / Bibliography: p. 114.

The primary topic of this thesis is the construction of explicit projective equations for the modular curves $X_0(N)$. The techniques may also be used to obtain equations for $X_0^+(p)$ and, more generally, $X_0(N) / W_n$. The thesis contains a number of tables of results. In particular, equations are given for all curves $X_0(N)$ having genus $2 le g le 5$. Equations are also given for all $X_0^+(p)$ having genus 2 or 3, and for the genus 4 and 5 curves $X_0^+(p)$ when $p le 251$. The most successful tool used to obtain these equations is the canonical embedding, combined with the fact that the differentials on a modular curve correspond to the weight 2 cusp forms. A second method, designed specifically for hyperelliptic curves, is given. A method for obtaining equations using weight 1 theta series is also described. Heights of modular curves are studied and a discussion is given of the size of coefficients occurring in equations for $X_0(N)$. Finally, the explicit equations are used to study the rational points on $X_0^+(p)$. Exceptional rational points on $X_0^+(p)$ are exhibited for $p = 73,103,137$ and 191.

This thesis is concerned with the Langlands program; namely the global Langlands correspondence, Langlands functoriality, and a conjecture of Gross. In chapter 1, we cover the most important background material needed for this thesis. This includes material on reductive groups and their root data, the definition of automorphic representations and a general overview of the Langlands program, and Gross' conjecture concerning attaching l-adic Galois representations to automorphic representations on certain reductive groups G over ℚ. In chapter 2, we show that odd-dimensional definite unitary groups satisfy the hypotheses of Gross' conjecture and verify the conjecture in this case using known constructions of automorphic l-adic Galois representations. We do this by verifying a specific case of a generalisation of Gross' conjecture; one should still get l-adic Galois representations if one removes one of his hypotheses but with the cost that their image lies in ^CG(ℚ_l) as opposed to ^LG(ℚ_l). Such Galois representations have been constructed for certain automorphic representations on G, a definite unitary group of arbitrary dimension, and there is a map ^CG(ℚ_l) → ^LG(ℚ_l) precisely when G is odd-dimensional. In chapter 3, which forms the main part of this thesis, we show that G = U_n(B) where B is a rational definite quaternion algebra satisfies the hypotheses of Gross' conjecture. We prove that one can transfer a cuspidal automorphic representation π of G to a π' on Sp_2n (a Jacquet-Langlands type transfer) provided it is Steinberg at some finite place. We also prove this when B is indefinite. One can then transfer π′ to an automorphic representaion of GL_2n+1 using the work of Arthur. Finally, one can attach l-adic Galois representations to these automorphic representations on GL_2n+1, provided we assume π is regular algebraic if B is indefinite, and show that they have orthogonal image.

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Made available in DSpace on 2014-12-17T15:05:00Z (GMT). No. of bitstreams: 1 SaraMSL_DISSERT.pdf: 4622900 bytes, checksum: 9c1229ad25c3c58fdb8c23fa644070ee (MD5) Previous issue date: 2013-01-22 / Conselho Nacional de Desenvolvimento Cient?fico e Tecnol?gico / Among the many methodological resources that the mathematics teacher can use in the classroom, we can cite the History of Mathematics which has contributed to the development of activities that promotes students curiosity about mathematics and its history. In this regard, the present dissertation aims to translate and analyze, mathematically and historically, the three works of Euler about amicable numbers that were writed during the Eighteenth century with the same title: De numeris amicabilibus. These works, despite being written in 1747 when Euler lived in Berlin, were published in different times and places. The first, published in 1747 in Nova Acta Eruditorum and which received the number E100 in the Enestr?m index, summarizes the historical context of amicable numbers, mentions the formula 2nxy & 2nz used by his precursors and presents a table containing thirty pairs of amicable numbers. The second work, E152, was published in 1750 in Opuscula varii argument. It is the result of a comprehensive review of Euler s research on amicable numbers which resulted in a catalog containing 61 pairs, a quantity which had never been achieved by any mathematician before Euler. Finally, the third work, E798, which was published in 1849 at the Opera postuma, was probably the first among the three works, to be written by Euler / Entre os diversos recursos metodol?gicos que podem ser trabalhados na sala de aula pelo professor de matem?tica, podemos citar a Hist?ria da Matem?tica que contribui para a elabora??o de atividades que promovam curiosidade hist?rica e matem?tica nos discentes. Assim, a presente disserta??o objetiva traduzir e analisar, matematicamente e historicamente, os tr?s trabalhos de Euler sobre os n?meros amig?veis que foram escritos durante o s?culo XVIII com o mesmo t?tulo: De numeris amicabilibus. Estes trabalhos, apesar de terem sido escritos em 1747 quando Euler vivia em Berlim, eles foram publicados em datas e lugares diferentes. O primeiro, publicado em 1747 na Nova Acta Eruditorum e que recebeu a numera??o E100 do ?ndex Enestr?m, apresenta resumidamente o contexto hist?rico dos n?meros amig?veis, menciona a f?rmula 2nxy & 2nz que foi usada por seus precursores e ainda apresenta uma tabela contendo como resultado trinta pares de n?meros amig?veis. Por sua vez, o segundo trabalho, E152, foi publicado em 1750 na Opuscula varii argumenti, ele ? o resultado de uma an?lise completa da pesquisa de Euler sobre os n?meros amig?veis que resultou em um cat?logo contendo 61 pares, quantidade n?o alcan?ada por nenhum matem?tico antes de Euler. Por fim, o terceiro trabalho, E798, que foi publicado em 1849 na Opera postuma, provavelmente tenha sido o primeiro, entre os tr?s trabalhos, a ser escrito por Euler