Portion of Thesis (Ph. D.)--Johns Hopkins University, 1932. / Vita. "Reprinted from ELH, a journal of English literary history, vol. I, no. 3, December, 1934"--P. 250.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1955. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves [207]-210).

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.

Texte abrégé de: Thèse--Lettres--Paris IV, 1981. / Bibliogr. p. 666-689. Index.

Not available / text

For a fixed rational prime p and primitive p-th root of unity ζ, we consider the Jacobian, J, of the complete non-singular curve give by equation yᵖ = xᵃ(1 − x)ᵇ. These curves are quotients of the p-th Fermat curve, given by equation xᵖ+yᵖ = 1, by a cyclic group of automorphisms. Let k = Q(ζ) and k(S) be the maximal extension of k unramified away from p inside a fixed algebraic closure of k. We produce a formula for the image of certain coboundary maps in group cohomology given in terms of Massey products, applicable in a general setting. Under specific circumstance, stated precisely below, we can use this formula and a pairing in the Galois cohomology of k(S) over k studied by W. McCallum and R. Sharifi in [MS02] to produce non-trivial elements in the Tate-Shafarevich group of J. In particular, we prove a theorem for predicting when the image of certain cyclotomic p-units in the Selmer group map non-trivially into X(k, J). / Q(zeta) and k_S be the maximal extension of k unramified away from p inside a fixed algebraic closure of k. We produce a formula for the image of certain coboundary maps in group cohomology given in terms of Massey products, applicable in a general setting. Under specific circumstance, stated precisely below, we can use this formula and a pairing in the Galois cohomology of k_S over k studied by W. McCallum and R. Sharifi to produce non-trivial elements in the Tate-Shafarevich group of J. In particular, we prove a theorem for predicting when the image of certain cyclotomic p-units in the Selmer group map non-trivially into Shah(k,J).

Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisor: Felipe Voloch. Vita. Includes bibliographical references. Also available from UMI.

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.

Bei elliptischen Kurven E/K über einem Zahlkörper K zwingt die Cassels-Tate Paarung die Ordnung der Tate-Shafarevich Gruppe Sha(E/K) zu einem Quadrat. Ist A/K eine prinzipal polarisierte abelschen Varietät, so ist bewiesen, daß die Ordnung von Sha(A/K) ein Quadrat oder zweimal ein Quadrat ist. William Stein vermutet, daß es für jede quadratfreie positive ganze Zahl k eine abelsche Varietät A/Q gibt, mit #Sha(A/Q)=kn². Jedoch ist es ein offenes Problem was zu erwarten ist, wenn die Dimension von A/Q beschränkt wird. Betrachtet man ausschließlich abelsche Flächen B/Q, so liefern Resultate von Poonen, Stoll und Stein Beispiele mit #Sha(B/Q)=kn², für k aus {1,2,3}. Diese Arbeit studiert tiefgehend nicht-einfache abelsche Flächen B/Q, d.h. es gibt elliptische Kurven E_1/Q und E_2/Q und eine Isogenie phi: E_1 x E_2 -> B. Relativ zur quadratischen Ordnung der Tate-Shafarevich Gruppe von E_1 x E_2 soll die Ordnung von Sha(B/Q) bestimmt werden. Um dieses Ziel zu erreichen wird die Isogenie-Invarianz der Vermutung von Birch und Swinnerton-Dyer ausgenutzt. Für jedes k aus {1,2,3,5,6,7,10,13,14} wird eine nicht-einfache, nicht-prinzipal polarisierte abelsche Fläche B/Q konstruiert, mit #Sha(B/Q)=kn². Desweiteren wird computergestützt berechnet wie oft #Sha(B/Q)=5n², sofern die Isogenie phi: E_1 x E_2 -> B zyklisch vom Grad 5 ist. Es stellt sich heraus, daß dies bei circa 50% der ersten 20 Millionen Beispielen der Fall ist. Abschließend wird gezeigt, daß wenn phi: E_1 x E_2 -> B zyklisch ist und #Sha(B/Q)=kn², so liegt k in {1,2,3,5,6,7,10,13}. Bei allgemeinen Isogenien phi: E_1 x E_2 -> B bleibt es unklar, ob k nur endlich viele verschiedene Werte annehmen kann. Im Anhang wird auf abelsche Flächen eingegangen, welche isogen zu der Jacobischen J einer hyperelliptischen Kurve über Q sind. Mit den in dieser Arbeit entwickelten Techniken können, anhand gewisser zyklischer Isogenien phi: J -> B, für jedes k in {11,17,23,29} Beispiele mit #Sha(B/Q)=kn² gegeben werden. / For elliptic curves E/K over a number field K the Cassels-Tate pairing forces the order of the Tate-Shafarevich group Sha(E/K) to be a perfect square. It is known, that if A/K is a principally polarised abelian variety, then the order of Sha(A/K) is a square or twice a square. William Stein conjectures that for any given square-free positive integer k there is an abelian variety A/Q, such that #Sha(A/Q)=kn². However, it is an open question what to expect if the dimension of A/Q is bounded. Restricting to abelian surfaces B/Q, then results of Poonen, Stoll and Stein imply that there are examples such that #Sha(B/Q)=kn², for k in {1,2,3}. In this thesis we focus in depth on non-simple abelian surfaces B/Q, i.e. there are elliptic curves E_1/Q and E_2/Q and an isogeny phi: E_1 x E_2 -> B. We want to compute the order of Sha(B/Q) with respect to the order of the Tate-Shafarevich group of E_1 x E_2, which has square order. To achieve this goal, we explore the invariance under isogeny of the Birch and Swinnerton-Dyer conjecture. For each k in {1,2,3,5,6,7,10,13,14} we construct a non-simple non-principally polarised abelian surface B/Q, such that #Sha(B/Q)=kn². Furthermore, we compute numerically how often the order of Sha(B/Q) equals five times a square, for cyclic isogenies phi: E_1 x E_2 -> B of degree 5. It turns out that this happens to be the case in approx. 50% of the first 20 million examples we have checked. Finally, we prove that if there is a cyclic isogeny phi: E_1 x E_2 -> B and #Sha(B/Q)=kn², then k is in {1,2,3,5,6,7,10,13}. For general isogenies phi: E_1 x E_2 -> B it remains unclear, whether there are only finitely many possibilities for k. In the appendix, we briefly consider abelian surfaces B/Q being isogenous to Jacobians J of hyperelliptic curves over Q. The techniques developed in this thesis allow to understand certain cyclic isogenies phi: J -> B. For each k in {11,17,23,29}, we provide an example with #Sha(B/Q)=kn².

There is a thread of darkness that seems to run through much of the canon of U.S. authors. There are, at the heart of us all, the questions we ask ourselves about who we are and what we mean to ourselves and others and to the places where we have lived. I believe that most of the body of writings produced in this country attempt to answer these questions in some form. Allen Tate wrote The Fathers in 1932, nearly seventy years after the Civil War, or the War Between the States. Perhaps one of the most critical moments in the process of how we became modern Americans, this period of history still resonates within our understanding. Tate, who was a Virginian and a Southerner, sought to understand what the South was and what it meant to modern America. The South became Tate's literary construct, a construct that included the abyss he would have to search. My belief is that Tate's South is an abyss which contains the answers to our questions of identity. The Fathers deals with identity through family and social structures in a changing South. Many may not be familiar with the world of the Civil War South that Tate was examining. Tate shows that depths of blackness can be found in the institutions of humans as well as in the natural world.