Thesis (doctoral)--Grossherzoglich Hessische Ludwigs-Universität zu Giessen, 1912. / "Sonderabdruck aus dem 27. Bande der "Mathematischen und Naturwissenschaftlichen Berichte aus Ungarn"--T.p. verso. Vita. Includes bibliographical references.

The purpose of this thesis is to construct nontrivial elements in the Bloch--Kato Selmer group of the symmetric cube of the Galois representation attached to a cuspidal holomorphic eigenform 𝐹 of level 1. The existence of such elements is predicted by the Bloch--Kato conjecture. This construction is carried out under certain standard conjectures related to Langlands functoriality. The broad method used to construct these elements is the one pioneered by Skinner and Urban in [SU06a] and [SU06b]. The construction has three steps, corresponding to the three chapters of this thesis. The first step is to use parabolic induction to construct a functorial lift of 𝐹 to an automorphic representation π of the exceptional group G₂ and then locate every instance of this functorial lift in the cohomology of G₂. In Eisenstein cohomology, this is done using the decomposition of Franke--Schwermer [FS98]. In cuspidal cohomology, this is done assuming Arthur's conjectures in order to classify certain CAP representations of G₂ which are nearly equivalent to π, and also using the work of Adams--Johnson [AJ87] to describe the Archimedean components of these CAP representations. This step works for 𝐹 of any level, even weight 𝑘 ≥ 4, and trivial nebentypus, as long as the symmetric cube 𝐿-function of 𝐹 vanishes at its central value. This last hypothesis is necessary because only then will the Bloch--Kato conjecture predict the existence of nontrivial elements in the symmetric cube Bloch--Kato Selmer group. Here this hypothesis is used in the case of Eisenstein cohomology to show the holomorphicity of certain Eisenstein series via the Langlands--Shahidi method, and in the case of cuspidal cohomology it is used to ensure that relevant discrete representations classified by Arthur's conjecture are cuspidal and not residual. The second step is to use the knowledge obtained in the first step to 𝓅-adically deform a certain critical 𝓅-stabilization 𝜎π of π in a generically cuspidal family of automorphic representations of G₂. This is done using the machinery of Urban's eigenvariety [Urb11]. This machinery operates on the multiplicities of automorphic representations in certain cohomology groups; in particular, it can relate the location of π in cohomology to the location of 𝜎π in an overconvergent analogue of cohomology and, under favorable circumstances, use this information to 𝓅-adically deform 𝜎π in a generically cuspidal family. We show that these circumstances are indeed favorable when the sign of the symmetric functional equation for 𝐹 is -1 either under certain conditions on the slope of 𝜎π, or in general when 𝐹 has level 1. The third and final step is to, under the assumption of a global Langlands correspondence for cohomological automorphic representations of G₂, carry over to the Galois side the generically cuspidal family of automorphic representations obtained in the second step to obtain a family of Galois representations which factors through G₂ and which specializes to the Galois representation attached to π. We then show this family is generically irreducible and make a Ribet-style construction of a particular lattice in this family. Specializing this lattice at the point corresponding to π gives a three step reducible Galois representation into GL₇, which we show must factor through, not only G₂, but a certain parabolic subgroup of G₂. Using this, we are able to construct the desired element of the symmetric cube Bloch--Kato Selmer group as an extension appearing in this reducible representation. The fact that this representation factors through the aforementioned parabolic subgroup of G₂ puts restrictions on the extension we obtain and guarantees that it lands in the symmetric cube Selmer group and not the Selmer group of 𝐹 itself. This step uses that 𝐹 is level 1 to control ramification at places different from 𝓅, and to ensure that 𝐹 is not CM so as to guarantee that the Galois representation attached to π has three irreducible pieces instead of four.

The work of mathematical giants, such as Lobachevsky, Gauss, Riemann, Klein and Poincaré, to name a few, lies at the foundation of the study of the highly structured Riemann surfaces, which allow definition of holomorphic maps, corresponding to analytic maps in the theory of complex analysis. A topological result of Poincaré states that every path-connected Riemann surface can be realised by a construction of identifying congruent points in the complex plane, the Riemann sphere or the hyperbolic plane; just three simply connected surfaces that cover the underlying Riemann surface. This requires the discontinuous action of a discrete subgroup of the automorphisms of the corresponding space. In the hyperbolic plane, which is the richest source for Riemann surfaces, these groups are called Fuchsian, and there are several ways to study the action of such groups geometrically by computing fundamental domains. What is accomplished in this thesis is a combination of the methods found by Reidemeister & Schreier, Singerman and Voight, and thus provides a unified way of finding Dirichlet domains for subgroups of cofinite groups with a given index. Several examples are considered in-depth.

Submitted by Laura Rezzieri Gambera null (laura.rgambera@gmail.com) on 2016-04-14T18:58:22Z No. of bitstreams: 1 Dissertação de mestrado - Laura R Gambera.pdf: 953384 bytes, checksum: a1e3addb576bbd6acb2de9e0d0d09394 (MD5) / Approved for entry into archive by Felipe Augusto Arakaki (arakaki@reitoria.unesp.br) on 2016-04-18T13:27:01Z (GMT) No. of bitstreams: 1 gambera_lr_me_sjrp.pdf: 953384 bytes, checksum: a1e3addb576bbd6acb2de9e0d0d09394 (MD5) / Made available in DSpace on 2016-04-18T13:27:01Z (GMT). No. of bitstreams: 1 gambera_lr_me_sjrp.pdf: 953384 bytes, checksum: a1e3addb576bbd6acb2de9e0d0d09394 (MD5) Previous issue date: 2016-03-29 / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Neste trabalho estudamos a existência de solução fraca quase automórfica para equações diferenciais abstratas de segunda ordem descritas na forma x’’(t) = Ax(t) + f(t, x(t)), t real, onde x(t) pertence a X para todo t real, X é um espaço de Banach, A : D(A) C X -> X é o gerador infinitesimal de uma família cosseno fortemente contínua de operadores lineares limitados em X e f : R x X -> X é uma função apropriada. / In this work we study the existence of an almost automorphic mild solution to second order abstract differential equations given by x’’(t) = Ax(t) + f(t, x(t)), t real, where x(t) lies in X for all t real, X is a Banach space, A : D(A) C X ->X is the infinitesimal generator of a strongly continuous cosine family of bounded linear operators on X and f : R x X -> X is an appropriate function. / FAPESP: 2013/22813-3

Dans cette thèse, nous affinons l'étude des fonctions presque automorphes sur time scales introduites dans la littérature par Lizama et Mesquita, nous explorons de nouvelles propriétés de ces fonctions et appliquons les résultats à étudier l'existence et l'unicité de solution presque automorphe d'une nouvelle classe d'équations dynamiques sur time scales. Puis nous introduisons la notion de fonction presque automorphe de classe Cn, nous investiguons les propriétés fondamentales de ces fonctions et utilisons les résultats pour établir l'existence, l'unicité et la stabilité globale et exponentielle de solution presque automorphe de classe C1 d'un système d'équations dynamiques avec délai variable fini modélisant un réseau de neurones. Ensuite nous présentons le concept de fonctions asymptotiquement presque automorphes de classe Cn. Nous démontrons quasiment toutes les propriétés de ces fonctions, lesquelles nous permettent, sous des hypothèses convenables, d'établir, d'une part, que l'unique solution d'un problème avec condition initiale est asymptotiquement presque automorphe de classe C1, et d'autre part, l'existence et l'unicité de solution asymptotiquement presque automorphe pour une équation intégro-dynamque avec conditon initiale non locale sur time scales. Enfin, en utilisant la notion de semi-groupe sur time scales de Hamza et Oraby, nous généralisons les résultats de Lizama et Mesquita en dimension infinie, c'est-à-dire, nous étudions l'existence et l'unicité des solutions presque automorphes pour des équations dynamiques semi linéaires abstraites sur time scales. / In this thesis, we refine the notion of almost automorphic functions on time scales introduced in the literature by Lizama and Mesquita, we explore some new properties of such functions and apply the results to study the existence and uniqueness of almost automorphic solution for a new class of dynamic equations on time scales. Then we introduce the concept of almost automorphic functions of order n on time scales, we investigate the fundamental properties of these functions and we use the findings to establish the existence and uniqueness and the global stability of almost automorphic solution of one to a first order dynamical equation with finite time varying delay. Then we present the concept of asymptotically almost automorphic functions of order n on time scales. We study the properties of these functions and we use the results to prove, under suitable hypothesis, that the unique solution to a problem with initial condition is asymptotically almost automorphic of order one at the one hand, and the existence and uniqueness of asymptotically almost automorphic solution for an integro-dynamic equation with nonlocal initial conditon on time scales in other hand. Finally, using the concept of semigroup on time scales introduced by Hamza and Oraby, we generalize the results in Lizama and Mesquita's paper for abstract Banach spaces, that is, we study the existence and uniqueness of almost automorphic solution for semilinear abstract dynamic equations on time scales.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we first develop a brief theoretical approach of semigroups of bounded linear operators, culminating on Hille-Yosida Theorem. Then we used the extrapolation theory to study su cient conditions to obtain existence and uniqueness of Almost Automorphic and Pseudo-Almost Automorphic mild solutions, through the Banach's Fixed Point Theorem for the semilinear evolution equation x(t) = Ax(t) + f(t; x(t)); t E R, where A : D(A) X ! X is a Hille-Yosida operator of negative type and not necessary dense domain on the Banach space X. / Neste trabalho, desenvolvemos inicialmente uma breve abordagem te orica dos semigrupos de operadores lineares limitados, culminando no Teorema de Hille-Yosida. Em seguida, usamos a teoria de extrapolação a fim de estudar condições suficientes para obtermos a existência e a unicidade de soluções brandas Quase Automórficas e Pseudo-quase Automórficas, por meio do Teorema do Ponto Fixo de Banach, para a equação de evolução semilinear x(t) = Ax(t) + f(t; x(t)); t E R, onde A : D(A) X ! X é um operador de Hille-Yosida de tipo negativo e dom ínio não necessariamente denso, definido no espaço de Banach X.