The central theme for this paper is provided by the following three statements: (1) Every compact connected 1-manifold is S1. (2) Every compact connected simply connected 2-manifold is S2. (3) Every compact connected simply connected 3-manifold is S3. We provide proofs of statements (1) and (2). The veracity of the third statement, the Poincaré Conjecture, has not been determined. It is known that should a counter-example exist it can be found by removing from S3 a finite collect ion of solid tori and sewing them back differently. We show that it is not possible to find a counterexample by removing from S3 a single solid torus of twist knot type or torus knot type and sewing it back differently. We treat as special cases a solid torus of trivial knot type and trefoil knot type.

Le théorème de Mordell-Weil affirme que, pour toute variété abélienne définie sur un corps de nombres, le groupe des points K-rationnels est de type fini. Plus exactement, ce groupe peut être vu comme le produit d’un groupe libre et d’un sous-groupe fini de points de torsion définis sur K. Il est naturel de se demander si l’on peut obtenir une borne uniforme pour le cardinal du sous-groupe fini des points de torsion définis sur une extension finie de K, dépendant uniquement du degré de cette extension, lorsque la variété abélienne varie. Pour ce qui est des courbes elliptiques définies sur un corps de nombres, Merel a prouvé en 1994 que l’on peut obtenir une borne uniforme en utilisant des méthodes développées par Mazur, Kenku-Momose et Kamienny. Cependant, il est aussi naturel de se demander si l’on peut obtenir une borne de ce cardinal, qui dépend uniquement du degré de cette extension,lorsque l’extension varie et la variété abélienne est fixée. Concernant cette dernière question Hindry et Ratazzi ont énoncé plusieurs résultats concernant certaines classes de variétés abéliennes. L’objectif de cette thèse, sera de présenter des nouveaux résultats dans cette direction. On se concentrera sur la classe de variétés abéliennes de type III pleinement de type Lefschetz, c’est-à-dire, telles que leur groupe de Mumford-Tate soit le groupe des similitudes orthogonales qui commutent avec les endomorphismes et telles qu’elles vérifient la conjecture de Mumford-Tate. On démontre des nouveaux résultats concernant la conjecture de Mumford-Tate. En particulier, on fournit une liste de variétés abéliennes dont on sait prouver qu’elles sont pleinement de type Lefschetz. / Mordell-Weil’s theorem states that, for an abelian variety defined over a number field K the group of K-rational points is finitely generated. More precisely, it can be seen as a product of a free group by a finite subgroup of torsion points over K. One can wonder if we can get an uniform bound for the order of the subgroup of torsion points over a finite extension L over K, depending on the degree of this extension and the dimension of the abelian variety, when the abelian variety varies in a certain class. For elliptic curves defined over a number field K, Merel proved in 1994 that we can get a uniform bound using methods developed by Mazur, Kenku-Momose and Kamienny. A complementary question would be to ask if we can get a bound for the order of the subgroup of torsion points over a finite extension L over K, depending on the degree of this extension and the dimension of the abelian variety, when L varies over all the finite extensions of K and the abelian variety is fixed. This question had been already answered by Hindry and Ratazzi for certain classes of abelian variety.This thesis will focus on this last question and will extend the previous results. We are going to present some new results concerning the class of abelian variety of type III in Albert’s classification and “fully of Lefschetz type” (i.e. whose Mumford-Tate group is the group of symplectic or orthogonal similitudes commuting with endomorphisms and which satisfy the Mumford-Tate conjecture). We also show some new results in the direction of the Mumford-Tate conjecture. Moreover, we present a list of abelian varieties which, we know, are fully of Lefschetz type.

Dans le premier chapitre de cette thèse on rappelle l'énoncé ainsi que des équivalents de la conjecture de Leopoldt puis l'on donne un algorithme permettant de vérifier cette conjecture pour un corps de nombre et premier donnés. Pour la suite on suppose cette conjecture vraie pour le premier p fixé Et on étudie la torsion du groupe de Galois de l'extension abélienne maximale p-ramifiée. On présente une méthode qui détermine effectivement les facteurs invariants de ce groupe fini. Dans le troisième chapitre on donne des résultats numériques que l'on interpréte via des heuristiques à la Cohen-Lenstra. Dans le quatrième chapitre, à l'aide de l'algorithme qui permet le calcul de ce module, on donne des exemples de corps et de premiers vérifiant la conjecture de Greenberg. / In the first chapter of this thesis we explain Leopoldt's conjecture and some equivalent formulations. Then we give an algorithm that checks this conjecture for a given prime p and a number field. Next we assume that this conjecture is true, and we study the torsion part of the Galois group of the maximal abelian p-ramified p-extension of a given number field. We present a method to compute the invariant factors of this finite group. In the third chapter we give an interpretation of our numrical result by heuristics “à la” Cohen-Lenstra. In the fourth and last chapter, using our algorithm which computes this torsion submodule, we give new examples of numbers fields which satisfy Greenberg's conjecture.

Given a finite group G of cardinality N, the group determinant [theta]G associated to G is a homogeneous polynomial in N variables of degree N. We study two properties of [theta]G. First we determine the stabilizer of [theta](G) under the action of permuting its variables. Then we also prove that the Lehmer's constant for any finite abelian group must satisfy a system of congruence equations. In particular when G is a p-group, we can strengthen the result to establish upper and lower bounds for the Lehmer's constant. / text

The main theme in this thesis is the ABC conjecture. We prove some partial results towards it and we find new applications of this conjecture, mainly in the context of B\"uchi's n squares problem (which has consequences in logic related to Hilbert's tenth problem) and squarefree values of polynomials. We also study related topics, such as arithmetic properties of additive subgroups of Hecke algebras, function field and meromorphic value distribution, and undecidability of the positive existential theories over languages of arithmetic interest. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2014-04-28 10:47:54.064

Master of Science / Department of Mathematics / Christopher Pinner / In 1988, Masser and Oesterlé conjectured that if A,B,C are co-prime integers satisfying A + B = C, then for any ε > 0, max{|A|,|B|,|C|}≤ K(ε)Rad(ABC)[superscript]1+ε, where Rad(n) denotes the product of the distinct primes dividing n. This is known as the ABC Conjecture. Versions with the ε dependence made explicit have also been conjectured. For example in 2004 A. Baker suggested that max{|A|,|B|,|C|}≤6/5Rad(ABC) (logRad(ABC))ω [over] ω! where ω = ω(ABC), denotes the number of distinct primes dividing A, B, and C. For example this would lead to max{|A|,|B|,|C|} < Rad(ABC)[superscript]7/4. The ABC Conjecture really is deep. Its truth would have a wide variety of applications to many different aspects in Number Theory, which we will see in this report. These include Fermat’s Last Theorem, Wieferich Primes, gaps between primes, Erdős-Woods Conjecture, Roth’s Theorem, Mordell’s Conjecture/Faltings’ Theorem, and Baker’s Theorem to name a few. For instance, it could be used to prove Fermat’s Last Theorem in only a couple of lines. That is truly fascinating in the world of Number Theory because it took over 300 years before Andrew Wiles came up with a lengthy proof of Fermat’s Last Theorem. We are far from proving this conjecture. The best we can do is Stewart and Yu’s 2001 result max{log|A|,log|B|,log|C|}≤ K(ε)Rad(ABC)[superscript]1/3+ε. (1) However, a polynomial version was proved by Mason in 1982.

Some heuristic arguments are given in support of the following conjecture: If the Riemann Hypothesis (RH) does not hold then the number of zeroes of the Riemann zeta function with real part σ >  ½ is infinite.

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