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 study of periods of automorphic forms using the theta correspondence and the Weil representation was initiated by Waldspurger and his work relating Fourier coefficients of modular forms of half-integral weight, periods over tori of modular forms of integral weight and special values of L-functions attached to these modular forms. In this thesis, we show that there are general relations among periods of automorphic forms on groups related by the theta correspondence. For example, if G is a symplectic group and H is an orthogonal group over a number field k, these relations are identities equating Fourier coefficients of cuspidal automorphic forms on G (relative to the Siegel parabolic subgroup) and periods of cuspidal automorphic forms on H over orthogonal subgroups. These identities are quite formal and follow from the basic properties of theta functions and the Weil representation; further study is required to show how they compare to the results of Waldspurger. The second part of this thesis shows that, under some restrictions, the identities alluded to above are the result of a comparison of nonstandard relative traces formulas. In this comparison, the relative trace formula for H is standard however the relative trace formula for G is novel in that it involves the trace of an operator built from theta functions. The final part of this thesis explores some preliminary results on local height pairings of special cycles on the p-adic upper half plane following the work of Kudla and Rapoport. These calculations should appear as the local factors of arithmetic orbital integrals in an arithmetic relative trace formula built from arithmetic theta functions as in the work of Kudla, Rapoport and Yang. Further study is required to use this approach to relate Fourier coefficients of modular forms of half-integral weight and arithmetic degrees of cycles on Shimura curves (which are the analogues in the arithmetic situation of the periods of automorphic forms over orthogonal subgroups).

The study of periods of automorphic forms using the theta correspondence and the Weil representation was initiated by Waldspurger and his work relating Fourier coefficients of modular forms of half-integral weight, periods over tori of modular forms of integral weight and special values of L-functions attached to these modular forms. In this thesis, we show that there are general relations among periods of automorphic forms on groups related by the theta correspondence. For example, if G is a symplectic group and H is an orthogonal group over a number field k, these relations are identities equating Fourier coefficients of cuspidal automorphic forms on G (relative to the Siegel parabolic subgroup) and periods of cuspidal automorphic forms on H over orthogonal subgroups. These identities are quite formal and follow from the basic properties of theta functions and the Weil representation; further study is required to show how they compare to the results of Waldspurger. The second part of this thesis shows that, under some restrictions, the identities alluded to above are the result of a comparison of nonstandard relative traces formulas. In this comparison, the relative trace formula for H is standard however the relative trace formula for G is novel in that it involves the trace of an operator built from theta functions. The final part of this thesis explores some preliminary results on local height pairings of special cycles on the p-adic upper half plane following the work of Kudla and Rapoport. These calculations should appear as the local factors of arithmetic orbital integrals in an arithmetic relative trace formula built from arithmetic theta functions as in the work of Kudla, Rapoport and Yang. Further study is required to use this approach to relate Fourier coefficients of modular forms of half-integral weight and arithmetic degrees of cycles on Shimura curves (which are the analogues in the arithmetic situation of the periods of automorphic forms over orthogonal subgroups).

Thesis (doctoral)--Universität Bonn, 1989. / Bibliography: p. 71-72.

Thesis--Rijksunivers teit te Utrecht. / Includes bibliographical references (p. 103-104).

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.

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Résumé anglais: This thesis consists of an introductory text followed by two separate parts which may be read independently of each other. In Part I we analyze certain hyperbolic structures arising when studying gravity in the vicinity of spacelike singularities (the BKL-limit). In this limit, spatial points decouple and the dynamics exhibits ultralocal behaviour which may be mapped to an auxiliary problem given in terms of a (possibly chaotic) hyperbolic billiard. In all supergravities arising as low-energy limits of string theory or M-theory, the billiard dynamics takes place within the fundamental Weyl chambers of certain hyperbolic Kac-Moody algebras, suggesting that these algebras generate hidden infinite-dimensional symmetries of gravity. We investigate the modification of the billiard dynamics when the original gravitational theory is formulated on a compact spatial manifold of arbitrary topology, revealing fascinating mathematical structures known as galleries. We further use the conjectured hyperbolic symmetry E10 to generate and classify certain cosmological (S-brane) solutions in eleven-dimensional supergravity. Finally, we show in detail that eleven-dimensional supergravity and massive type IIA supergravity are dynamically unified within the framework of a geodesic sigma model for a particle moving on the infinite-dimensional coset space E10/K(E10). Part II of the thesis is devoted to a study of how (U-)dualities in string theory provide powerful constraints on perturbative and non-perturbative quantum corrections. These dualities are typically given by certain arithmetic groups G(Z) which are conjectured to be preserved in the effective action. The exact couplings are given by moduli-dependent functions which are manifestly invariant under G(Z), known as automorphic forms. We discuss in detail various methods of constructing automorphic forms, with particular emphasis on a special class of functions known as (non-holomorphic) Eisenstein series. We provide detailed examples for the physically relevant cases of SL(2,Z) and SL(3,Z), for which we construct their respective Eisenstein series and compute their (non-abelian) Fourier expansions. We also discuss the possibility that certain generalized Eisenstein series, which are covariant under the maximal compact subgroup K(G), could play a role in determining the exact effective action for toroidally compactified higher derivative corrections. Finally, we propose that in the case of rigid Calabi-Yau compactifications in type IIA string theory, the exact universal hypermultiplet moduli space exhibits a quantum duality group given by the emph{Picard modular group} SU(2,1;Z[i]). To verify this proposal we construct an SU(2,1;Z[i])-invariant Eisenstein series, and we present preliminary results for its Fourier expansion which reveals the expected contributions from D2-brane and NS5-brane instantons. / Résumé francais: Cette thèse est composée d'une introduction suivie de deux parties qui peuvent être lues indépendemment. Dans la première partie, nous analysons des structures hyperboliques apparaissant dans l'étude de la gravité au voisinage d'une singularité de type espace (la limite BKL). Dans cette limite, les points spatiaux se découplent et la dynamique suit un comportement ultralocal qui peut être reformulé en termes d'un billiard hyperbolique (qui peut être chaotique). Dans toutes les supergravités qui sont des limites de basse énergie de théories de cordes ou de la théorie M, la dynamique du billiard prend place à l'intérieur des chambres de Weyl fondamentales de certaines algèbres de Kac-Moody hyperboliques, ce qui suggère que ces algèbres correspondent à des symétries cachées de dimension infinie de la gravité. Nous examinons comment la dynamique du billard est modifiée quand la théorie de gravité originale est formulée sur une variété spatiale compacte de topologie arbitraire, révélant ainsi de fascinantes structures mathématiques appelées galleries. De plus, dans le cadre de la supergravité à onze dimensions, nous utilisons la symétrie hyperbolique conjecturée E10 pour engendrer et classifier certaines solutions cosmologiques (S-branes). Finalement, nous montrons en détail que la supergravité à onze dimensions et la supergravité de type IIA massive sont dynamiquement unifiées dans le contexte d'un modèle sigma géodesique pour une particule se déplaçant sur l'espace quotient de dimension infinie E10/K(E10). La deuxième partie de cette thèse est consacrée à étudier comment les dualités U en théorie des cordes fournissent des contraintes puissantes sur les corrections quantiques perturbatives et non perturbatives. Ces dualités sont typiquement données par des groupes arithmétiques G(Z) dont il est conjecturé qu'ils préservent l'action effective. Les couplages exacts sont donnés par des fonctions des moduli qui sont manifestement invariantes sous G(Z), et qu'on appelle des formes automorphiques. Nous discutons en détail différentes méthodes de construction de ces formes automorphiques, en insistant particulièrement sur une classe spéciale de fonctions appelées séries d'Eisenstein (non holomorphiques). Nous présentons comme exemples les cas de SL(2,Z) et SL(3,Z), qui sont physiquement pertinents. Nous construisons les séries d'Eisenstein correspondantes et leurs expansions de Fourier (non abéliennes). Nous discutons également la possibilité que certaines séries d'Eisenstein généralisées, qui sont covariantes sous le sous-groupe compact maximal, pourraient jouer un rôle dans la détermination des actions effectives exactes pour les théories incluant des corrections de dérivées supérieures compactifiées sur des tores.