Orientador: Jose Plinio de Oliveira Santos / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-06T00:16:28Z (GMT). No. of bitstreams: 1 Estrada_EduardoLuis_M.pdf: 1229842 bytes, checksum: 17b398d3ef955f687a933b77677a7113 (MD5) Previous issue date: 2006 / Resumo: Abordamos, de maneira elementar, as estruturas algébrica e topológica sobre a qual são construídas as formas modulares, objetos principais do nosso estudo. Após a definição de formas modulares, realizamos um estudo particular sobre duas funções específicas relacionadas à teoria dos números: h(t) e. u(t). Trata-se de um texto introdutório, no qual apresentamos diversos conceitos e resultados extremamente importantes da teoria, tais como as demonstrações de que as duas funções supracitadas são formas modulares e a apresentação de uma fórmula explícita para seus sistemas multiplicadores / Abstract: In an elemmentary way, we have dealt with the algebraic and topological structures in which the modular forms are constructed. After the definition of this important tool, we have made a particular study about two specific functions related to number theory: h(t) and u(t). It is an introductory text, in which we have presented many concepts and results extremely importants of the theory, such as the proofs of the fact that the two functions h(t) and u(t) are modular forms and the presentation of an exact formula for their multiplier systems / Mestrado / Analise Matematica / Mestre em Matemática

There now exists an abundant collection of conjectures and results, of various complexities, regarding the diophantine properties of Shimura varieties. Two central such statements are the Andre-Oort and Zilber-Pink Conjectures, the first of which is known in many cases, while the second is known in very few cases indeed. The motivating result for much of this document is the modular case of the Andre-Oort Conjecture, which is a theorem of Pila. It is most commonly viewed as a statement about the simplest kind of Shimura varieties, namely modular curves. Here, we tend instead to view it as a statement about the properties of the classical modular j-function. It states, given a complex algebraic variety V, that V contains only finitely many maximal special subvarieties, where a special variety is one which arises from the arithmetic behaviour of the j-function in a certain natural way. The central question of this thesis is the following: what happens if in such statements we replace the j-function with some other kind of modular function; one which is less well-behaved in one way or another? Such modular functions are naturally called nonclassical modular functions. This question, as we shall see, can be studied using techniques of o-minimality and point-counting, but some interesting new features arise and must be dealt with. After laying out some of the classical theory, we go on to describe two particular types of nonclassical modular function: almost holomorphic modular functions and quasimodular functions (which arise naturally from the derivatives of the j-function). We go on to prove some results about the diophantine properties of these functions, including several natural Andre-Oort-type theorems, then conclude by discussing some bigger-picture questions (such as the potential for nonclassical variants of, say, Zilber-Pink) and some directions for future research in this area.