Este trabalho consiste de duas partes. Na primeira, desenvolvemos uma teoria de Nielsen equivariante para raizes de G-aplicações $f:X\\to Y$ equivariantes entre G-espaços topológicos Hausdorff, conexos, normais, localmente conexos por caminhos e semilocalmente simplesmente conexos, onde G é um grupo topológico, Na segunda parte, estudamos a questão da realização do G-número de Nielsen de raizes quando este é zero. / This work consists of two parts. In the firs one, we develop an equivariant Nielsen root theory for G-maps. We consider equivariant maps $f:X\\to Y$ between Hausdorff, connected, normal, locally path connected and semilocally simply connected G-spaces, where G is a topological group. In the second part, we study the question of the realization of G-Nielsen root number when it is zero.

Let E be a one-parameter family of elliptic curves over a number field. It is natural to expect the average root number of the curves in the family to be zero. All known counterexamples to this folk conjecture occur for families obeying a certain degeneracy condition. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we show that any non-degenerate family E has average root number 0, provided that two classical arithmetical conjectures hold for two homogeneous polynomials with integral coefficients constructed explicitly in terms of E.<br />The first such conjecture -- commonly associated with Chowla -- asserts the equidistribution of the parity of the number of primes dividing the integers represented by a polynomial. We prove the conjecture for homogeneous polynomials of degree 3.<br />The second conjecture used states that any non-constant homogeneous polynomial yields to a square-free sieve. We sharpen the existing bounds on the known cases by a sieve refinement and a new approach combining height functions, sphere packings and sieve methods.

Este trabalho consiste de duas partes. Na primeira, desenvolvemos uma teoria de Nielsen equivariante para raizes de G-aplicações $f:X\\to Y$ equivariantes entre G-espaços topológicos Hausdorff, conexos, normais, localmente conexos por caminhos e semilocalmente simplesmente conexos, onde G é um grupo topológico, Na segunda parte, estudamos a questão da realização do G-número de Nielsen de raizes quando este é zero. / This work consists of two parts. In the firs one, we develop an equivariant Nielsen root theory for G-maps. We consider equivariant maps $f:X\\to Y$ between Hausdorff, connected, normal, locally path connected and semilocally simply connected G-spaces, where G is a topological group. In the second part, we study the question of the realization of G-Nielsen root number when it is zero.

Este texto é resultado de um estudo detalhado da teoria topológica de raízes para aplicações de complexos CW 2-dimensionais em superfícies fechadas (compactas e sem bordo). Diversas abordagens dos problemas envolvidos nesta teoria são apresentadas, algumas inclusive bastante diferenciadas com respeito aos parâmetros da teoria clássica / This text is the result of a detailed study of the topological root theory for maps from 2-dimensional CW complex into closed surfaces (compact and without boundary surfaces). Several approaches to the problems involved in this theory are presented, some of which are quite different with respect to the parameters of the classical theory

Este texto é resultado de um estudo detalhado da teoria topológica de raízes para aplicações de complexos CW 2-dimensionais em superfícies fechadas (compactas e sem bordo). Diversas abordagens dos problemas envolvidos nesta teoria são apresentadas, algumas inclusive bastante diferenciadas com respeito aos parâmetros da teoria clássica / This text is the result of a detailed study of the topological root theory for maps from 2-dimensional CW complex into closed surfaces (compact and without boundary surfaces). Several approaches to the problems involved in this theory are presented, some of which are quite different with respect to the parameters of the classical theory