Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Esta dissertaÃÃo apresenta resultados sobre coberturas de grupos por sub-grupos abelianos, subgrupos de Sylow e subgrupos normais. O Teorema de Neumann à indispensÃvel no estudo das coberturas por subgrupos. Apresentamos no apÃndice C uma prova elementar de um resultado muito importante nas coberturas p-Sylow. / The paper results on the Coverage groups by abelian subgroups, subgroups of Sylow and normal subgroups. We present in appendix C an elementary proof a very important result in the coverage p-Sylow.

A Conjectura de Bateman-Horn dá condições sobre uma família de polinômios com coeficientes inteiros $f_1(X),\\dots,f_k(X)$ para que hajam infinitos $n \\in \\N$ tais que $f_1(n),\\dots,f_k(n)$ sejam todos primos, e determina qual deve ser o comportamento assintótico de tais inteiros $n$. Neste texto, vamos estudar essa conjectura, assim como um método desenvolvido por Solomon W. Golomb que pode ser usado para demonstrá-la. Veremos que esse cálculo prova a Conjectura de Bateman-Horn a menos da troca de um limite com uma série infinita, que é o único passo ainda não provado desse método. Também estudaremos uma tentativa para solucionar esse problema por meio do uso de teoremas abelianos de regularidade, e provaremos que teoremas tão gerais não são suficientes para provar a troca do limite com a série. / Given a family of polynomials with integer coefficients $f_1(X),\\dots,f_k(X)$, one would like to answer the following question: does there exist infinitely many $n \\in \\N$ such that $f_1(n),\\dots,f_k(n)$ are all primes? Schinzel conjectured that if these polynomials satisfy certain simple conditions, then the answer to this question is affirmative. Assuming these conditions, Bateman and Horn proposed a formula for the asymptotic density of the integers $n \\in \\N$ such that $f_1(n),\\dots,f_k(n)$ are all primes. In this text, we shall study the Bateman-Horn Conjecture, as well as a method proposed by Solomon W. Golomb that may be used to prove this conjecture. We shall see that Golomb\'s $\\Lambda$-method would prove the Bateman-Horn Conjecture, except for a single unproved step, namely, the commutation of a limit with an infinite series.

Dados dois conjuntos compactos e convexos K1, K2 em R², queremos saber se existem f e h, dois homeomorfismos de T², homotópicos à identidade, que comutam, com levantamentos F e H, tais que K1 e K2 são os seus conjuntos de rotação, respectivamente. Neste trabalho, mostramos alguns casos onde isto não pode acontecer, assumindo restrições nos conjuntos de rotação. Além disso, introduzimos o conceito de conjunto de rotação para semigrupos Abelianos finitamente gerados por homeomorfismos homotópicos à identidade, mostrando um caso em que o semigrupo é anular. / Let K1, K2 in R² be two convex, compact sets. We would like to know if there are commuting homeomorphisms f and h of T², homotopic to the identity, with lifts F and H, such that K1 and K2 are their rotation sets, respectively. In this work, we proof some cases where it cannot happen, assuming some restrictions on rotation sets. Besides that, we introduce the concept of rotation set for Abelian semi-groups finitely generated by homeomorphisms homotopic to the identity, showing a case where the semi-group is annular.

Orientador: Marcos Benevenuto Jardim / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-24T02:07:51Z (GMT). No. of bitstreams: 1 Castilho_TiagoNunes_D.pdf: 20018125 bytes, checksum: 181cd44948098969af37059c2215917a (MD5) Previous issue date: 2013 / Resumo: Esta tese trata de variedades de Prym e de semigrupos de Weierstrass, ambos no contexto de recobrimentos duplos de curvas ramificados. A partir da descrição da variedade de Prym em termos de um conjunto de fibrações lineares do recobrimento, estuda-se a dualidade entre o lugar onde a aplicação de Gauss sobre o divisor Prym-Theta se degenera e o divisor de ramos do recobrimento duplo, em que provarse uma relação entre as fibras da aplicação de Gauss e os semigrupos de Weierstrass das ramificações do recobrimento / Abstract: ln this thesis we present results about Prym varieties and Weierstrass semigroups, both in the context of ramified double covers of curves. From the description of the Prym variety by a set of linear fibrations, we study the duality between the place where the Gauss map on the Prym-Theta divisor degenerates and the branch divisor of the double covering, in which we prove a relation between the fibers of the Gauss map and the Weierstrass semigroups of branched points of the double covering / Doutorado / Matematica / Doutor em Matemática

Les polynômes modulaires sont utilisés dans le calcul de graphes d’isogénies, le calcul des polynômes de classes ou le comptage du nombre de points d’une courbe elliptique, et sont donc fondamentaux pour la cryptographie basée sur les courbes elliptiques. Des polynômes analogues sur les surfaces abéliennes principalement polarisées ont été introduits par Régis Dupont en 2006, qui a également proposé un algorithme pour les calculer, et des résultats théoriques sur ces polynômes ont été donnés dans un article de Bröker–Lauter, en 2009. Mais les polynômes sont très gros et ils n’ont pu être calculés que pour l’exemple minimal p = 2. Dans cette thèse, nous poursuivons les travaux de Dupont et Bröker–Lauter en permettant de calculer des polynômes modulaires pour des invariants basés sur les thêta constantes, avec lesquels nous avons pu calculer les polynômes jusqu’à p = 7, tout en démontrant des propriétés de ces polynômes. Mais des exemples plus grands ne semblent pas envisageables. Ainsi, nous proposons une nouvelle définition des polynômes modulaires dans laquelle l’on se restreint aux surfaces abéliennes principalement polarisées qui ont multiplication réelle par l’ordre maximal d’un corps quadratique réel afin d’obtenir des polynômes plus petits. Nous présentons alors de nombreux exemples de polynômes et des résultats théoriques. / Modular polynomials on elliptic curves are a fundamental tool used for the computation of graph of isogenies, class polynomials or for point counting. Thus, they are fundamental for the elliptic curve cryptography. A generalization of these polynomials for principally polarized abelian surfaces has been introduced by Régis Dupont in 2006, who has also described an algorithm to compute them, while theoretical results can been found in an article of Bröker– Lauter of 2009. But these polynomials being really big, they have been computed only in the minimal case p = 2. In this thesis, we continue the work of Dupont and Bröker–Lauter by defining and giving theoretical results on modular polynomials with new invariants, based on theta constants. Using these invariants, we have been able to compute the polynomials until p = 7 but bigger examples look intractable. Thus we define a new kind of modular polynomials where we restrict on the surfaces having real multiplication by the maximal order of a real quadratic field. We present many examples and theoretical results.

The purpose of this project was to study the classical result in this direction discovered by O. Ore in 1938, as well as related theorems and corollaries. Ore's Theorem and its corollaries provide us with several results relating distributive lattices with cyclic groups.

Soient $K$ un corps de nombres et $A$ une variété abélienne sur $K$ dont nous notons $g$ la dimension. Pour tout premier $ell$, le module de Tate $ell$-adique de $A$ nous fournit une représentation $ell$-adique du groupe de Galois absolu de $K$, et c'est à l'image de ces représentations galoisiennes que l'on s'intéresse dans cette thèse.Pour de nombreuses classes de variétés abéliennes on possède une description de ces images à une erreur finie près : le premier but de ce travail est de quantifier explicitement cette erreur dans plusieurs cas différents. On parvient à résoudre complètement le problème pour une courbe elliptique sans multiplication complexe, ou plus généralement pour un produit de telles courbes elliptiques, et pour toute variété abélienne géométriquement simple admettant multiplication complexe. Pour d'autres classes de variétés abéliennes $A/K$ on obtient seulement une description de l'image de Galois pour tout premier $ell$ plus grand qu'une certaine borne (que l'on calcule explicitement, et qui est polynomiale en le degré de $K$ et en la hauteur de Faltings de $A$) : nous prouvons de tels résultats pour toute surface abélienne semistable et géométriquement simple et pour les variétés dites "de type $operatorname{GL}_2$''. On montre également un résultat semblable, mais un peu affaibli, pour de nombreuses variétés abéliennes de dimension impaire dont l'anneau des endomorphismes est réduit à $mathbb{Z}$.On s'intéresse ensuite à l'action de Galois sur des variétés abéliennes non simples, et on donne des conditions suffisantes pour que les représentations galoisiennes qui leur sont associées se décomposent elles-mêmes en produit. Finalement on étudie l'intersection entre les extensions cyclotomiques d'un corps de nombres $K$ et les corps engendrés par les points de torsion d'une variété abélienne sur $K$, et on établit des propriétés d'uniformité des degrés de ces intersections. / Let $K$ be a number field and $A$ be a $g$-dimensional abelian variety over $K$. For every prime $ell$, the $ell$-adic Tate module of $A$ gives rise to an $ell$-adic representation of the absolute Galois group of $K$; in this thesis we set out to study the images of the Galois representations arising in this way.For various classes of abelian varieties a description of these images is known up to finite error, and the first aim of this work is to explicitly quantify this error for a number of different cases. We provide a complete solution for the case of elliptic curves without complex multiplication (and more generally for products thereof) and for geometrically simple abelian varieties of CM type. For other classes of abelian varieties we can only describe the Galois image when the prime $ell$ is above a certain bound (which we compute explicitly in terms of $A$, and which is polynomial in $[K:mathbb{Q}]$ and in the Faltings height of $A$): we obtain such results for geometrically simple, semistable abelian surfaces and for "$operatorname{GL}_2$-type" varieties. We also prove similar (but slightly weaker) results for many abelian varieties of odd dimension with trivial endomorphism algebra.We then consider the Galois action on non-simple abelian varieties, and we give sufficient conditions for the associated Galois representations to decompose as a product.Finally, we investigate the structure of the intersection between the cyclotomic extensions of a number field $K$ and the fields generated by the torsion points of an abelian variety over $K$, proving a uniformity property for the degrees of such intersections.

A Schur ring is a subring of the group algebra with a basis that is formed by a partition of the group. These subrings were initially used to study finite permutation groups, and classifications of Schur rings over various finite groups have been studied. Here we investigate Schur rings over various infinite groups, including free groups. We classify Schur rings over the infinite cyclic group.

Let R be a domain, V a left R-module, and L a composition series of direct summands of V. Our main results show that if U is a stabilizer group of L containing the McLain-group associated with L , then U determines the chain (L,⊆) uniquely up to isomorphism or anti-isomorphism.

We show that any Borel action on a standard Borel space of a group which is topologically isomorphic to the sum of a countable abelian group with a countable sum of lines and circles induces an orbit equivalence relation which is hypersmooth. We also show that any Borel action of a second countable locally compact abelian group on a standard Borel space induces an orbit equivalence relation which is essentially hyperfinite, generalizing a result of Gao and Jackson for the countable abelian groups.