The principal theorem of this thesis is a theorem by Peano on the existence of a solution to a certain integral equation. The two primary notions underlying this theorem are uniform convergence and equi-continuity. Theorems related to these two topics are proved in Chapter II. In Chapter III we state and prove a classical existence and uniqueness theorem for an integral equation. In Chapter IV we consider the approximation on certain functions by means of elementary expressions involving "bent line" functions. The last chapter, Chapter V, is the proof of the theorem by Peano mentioned above. Also included in this chapter is an example in which the integral equation has more than one solution. The first chapter sets forth basic definitions and theorems with which the reader should be acquainted.

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.