Thesis (M.S.)--Boise State University, 2009. / Title from t.p. of PDF file (viewed June 15, 2010). Includes abstract. Includes bibliographical references (leaf 82).

This thesis consists of three chapters. Chapter 1 explains how the research problems considered in this thesis fit into the investigation of local-global principle in the diophantine geometry, as well as gives a unified sketch of the proofs of the two main results in this thesis. Chapter 2 establishes a similar conclusion to Theorem B of a paper by Poonen and Voloch in another settings. Chapter 3 relates to the object considered in the main result of Chapter 2 to an old conjecture proposed by Skolem and solves some cases of its analog. / text

Fermat, Pierre de, is perhaps the most famous number theorist who ever lived. Fermat's Last Theorem states that the equation xn + yn = zn has no non-zero integer solutions for x, y and z when n>2.

Le but de cette thèse est d'obtenir des versions totalement explicite de deux résultats fondamentales sur les revêtements de courbes algébriques: le Théorème d'existence de Riemann et le théorème de Chevalley-Weil. La motivation de notre travail sur le Théorème d'existence de Riemann réside dans le domaine de l'analyse diophantienne effective, lorsque la technique des revêtements est largement utilisé: trés souvent il arrive qu'on ne connait que le degré du revêtement et les points de ramification, et pour travailler avec le revêtement il faut en avoir une description efficace. Le théorème de Chevalley-Weil est également indispensable dans l'analyse diophantienne, car il permet de réduire un problème diophantien sur la variété V à celui sur le revêtement W, ce qui peut être plus simple à étudier. Dans la thèse on obtient une version du théorème de Chevalley-Weil en dimension 1, explicite en tous les paramètres et nettement meilleur que les versions précédentes. / The purpose of this thesis is to obtain totally explicit versions for two fundamental results about coverings of algebraic curves: the Riemann Existence Theorem and the Chevalley-Weil Theorem. The motivation behind our work about Riemann Existence Theorem lies in the field of effective Diophantine analysis, where the covering technique is widely used: it happens quite often that only the degree of the covering and the ramification points are known, and to work with the covering curve, one needs to have an effective description of it. The Chevalley-Weil theorem is also indispensable in the Diophantine analysis because it reduces a Diophantine problem on the variety V to that on the covering variety W, which can often be simpler to deal. In the thesis we obtain a version of the Chevalley-Weil theorem in dimension 1, explicit in all parameters and considerably sharper than the previous versions. / La tesi si propone di ottenere versioni totalmente esplicite di due risultati fondamentali riguardanti rivestimenti di curve algebriche: il teorema di esistenza di Riemann e il teorema di Chevalley-Weil. Le motivazioni del nostro lavoro sul teorema di esistenza di Riemann risiedono nella analisi diofantea effettiva, dove le tecniche di rivestimento sono ampiamente utilizzate: capita spesso di conoscere solo il grado e i punti di ramificazione di un rivestimento, e per lavorare con la curva e' necessario averne una descrizione esplicita. Il teorema di Chevalley-Weil e' altrettanto indispensabile in analisi diofantea poiche' riduce un problema diofanteo su una varieta' V a quello di un rivestimento W, dove spesso e' piu' facile lavorare. Nella tesi otteniamo una versione totalmente esplicita del teorema di Chevalley-Weil in dimensione 1, con stime molto migliori di quelle precedentemente conosciute.

The main purpose of this thesis is to apply the modular approach to Diophantine equations to study some Fermat-type equations of signature (r; r; p) with r >/= 5 a fixed prime and “p” varying. In particular, we will study equations of the form x(r) + y(r) = Cz(p), where C is an integer divisible only by primes “q” is non-identical to 1; 0 (mod “r”) and obtain explicit arithmetic results for “r” = 5, 7, 13. We start with equations of the form x(5) + y(5) = Cz(p). Firstly, we attach two Frey curves E; F defined over Q(square root 5) to putative solutions of the equation. Then by using the work of J. Quer on embedding problems and on abelian varieties attached to Q-curves we prove that the p-adic Galois representations attached to E, F can be extended to p-adic representations E), (F) of Gal(Q=Q). Finally, we apply Serre's conjecture to the residual representations  (E), (F) and using Siksek's multi-Frey technique we conclude that the initial solution can not exist. We also describe a general method for attacking infinitely many equations of the form x(r) + y(r) = Cz(p) for all r>/= 7. The method makes use of elliptic curves over totally real fields, modularity and irreducibility results for representations attached to elliptic curves and level lowering theorems for Hilbert modular forms. Indeed, for each fixed “r” we produce several Frey curves defined over K+, the maximal totally real subfield of Q(xi-r). Moreover, if “r” is of the form 6k + 1 we prove the existence of a Frey curve defined over K(0) the subfield of K(+) of degree k. We prove also an irreducibility result for the mod “p” representations attached to certain elliptic curves and a modularity statement for elliptic curves over totally real abelian number fields satisfying some local conditions at 3. Finally, for r = 7 and r = 13 we are able to compute the required spaces of (Hilbert) newforms and by applying our general methods we obtain explicit arithmetic results for equations of signature (7; 7; p) and (13; 13; p). We end by providing two more Frey k-curves (a generalization of Q-curve), where “k” is a certain subfield of K(+), when “r” is a fixed prime of the form 4m+1. / En esta tesis, utilizaremos el método modular para profundizar en el estudio de las ecuaciones de tipo (r; r; p) para r un primo fijado. Empezamos por utilizar la teoría de J. Quer sobre variedades abelianas asociadas con Q-curvas y embedding problems para producir dos curvas de Frey asociadas con hipotéticas soluciones de infinitas ecuaciones de tipo (5; 5; p). Después, utilizando la conjetura de Serre y el método multi-Frey de Siksek demostraremos que las hipotéticas soluciones no pueden existir. Describiremos también un método general que nos permite atacar un número infinito de ecuaciones de tipo (r; r; p) para cada primo “r” mayor o igual que 7. El método hace uso de curvas elípticas sobre cuerpos de números, teoremas de modularidad, teoremas de bajada de nivel y formas modulares de Hilbert. Además, para ecuaciones de tipo (7; 7; p) y (13; 13; p) calcularemos los espacios de formas modulares relevantes y demostraremos que una familia infinita de ecuaciones no admite cierto tipo de soluciones. Además, demostraremos un nuevo teorema de modularidad para curvas elípticas sobre cuerpos totalmente reales abelianos. Finalmente, para primos congruentes con 1 módulo 4 propondremos dos curvas de Frey más. Demostraremos que son “k-curves” (una generalización de Q-curva) y también que satisfacen las propiedades necesarias para que pueda ser útiles en la aplicación del método modular.