A new way of proof of Szegö-type theorems is presented. The idea of the proof is based on the construction of "almost" inverse operator to the finite section T_n(a) of a Toeplitz operator T(a), which is close to the inverse operator in the trace norm (these "almost" inverses are well-known). This way of proof gives the possibility to write another representation for the second constant E_f(a), and in the scalar case to receive a shorter representation. Another observation is that the convergence in these theorems is strongly dependent on the smoothness of the generating function a.

Ce travail de thèse a pour but de construire et d’étudier une fibration de Hitchin pour les groupes qui apparaît naturellement lorsque l’on essaie de géométriser la formule des traces. On commence par construire une telle fibration en utilisant le semi-groupe de Vinberg. Sur ce semi-groupe de Vinberg, on montre qu’il existe un certain morphisme « polynôme caractéristique » muni d’une section naturelle, de même que dans le cas des algèbres de Lie. On montre également que l’on peut construire un centralisateur régulier au-dessus de cette base des polynômes caractéristiques qui est un schéma en groupes commutatif et lisse.On s’intéresse alors à des variantes pour les groupes des fibres de Springer affines pour lesquelles on remarque que l’introduction du semi-groupe de Vinberg permet d’obtenir une condition d’intégralité analogue à celle de Kazhdan-Lusztig. Ces fibres de Springer affines sont des analogues locaux des fibres de Hitchin. On obtient alors une formule de dimension pour ces fibres.Dans un troisième temps, on s’intéresse à l’aspect global de cette fibration pour laquelle on donne une interprétation modulaire et sur laquelle on construit l’action d’un champ de Picard, issu du centralisateur régulier. L’espace total de cette fibration étant en général singulier, nous étudions son complexe d’intersection. Cet espace de Hitchin s’obtient naturellement comme l’intersection du champ de Hecke avec la diagonale du champ des G-torseurs et on démontre que sur un ouvert suffisamment gros de la base de Hitchin, le complexe d’intersection de l’espace de Hitchin s’obtient par restriction de celui du champ de Hecke corrrespondant.Enfin, dans la dernière partie de cette thèse, on établit un théorème du support dans le cas où l’espace total est singulier analogue à celui de Ngô et l’on démontre que, dans le cas de la fibration de Hitchin, les supports qui interviennent sont reliés aux strates endoscopiques. / This main goal of this work is to construct and study the properties of Hitchin fibration for groups which appears naturally when we try to geometrize the trace formula. We begin by constructing this fibration using the Vinberg’s semigroup. On this semigroup, we show that there exists a characteristic polynomial morphism equipped with a natural section, analog at the Kostant’s one in the case of Lie algebras. We also show that there exists on the base of characteristic polynomials a regular centralizer scheme, which is a smooth commutative group scheme.Then, we are interested in some variant of affine Springer fibers, for which we see that the Vinberg’s semigroup appears naturally to obtain an integrality condition analog to Kazhdan-Lusztig’s one. These affine Springer fibers are local incarnation of Hitchin fibers.In a third time, we go back to the global case and give a modular interpretation of this new Hitchin fibration on which we construct an action of a Picard stack, coming from the regular centralizer.The total space of this fibration, even on the generically regular semisimple locus will be singular and we want to understand his intersection complex. This space can be obtained as the intersection of the Hecke stack with the diagonal of the stack of G-bundles and we show that on a sufficiently big open subset of the Hitchin base, the intersection complex of the Hitchin’s space is the restriction of the corresponding intersection complex on the Hecke stack.Finally, in the last part of this work, we establish a support theorem in the case of a singular total space, generalizing Ngo’s theorem et we show that in the case of Hitchin fibration, the supports that appear are related to the endoscopic strata.

This thesis is devoted to the study of operators on discrete structures. The operators are supposed to be self-adjoint and obey a certain translation invariance property. The discrete structures are given as Cayley graphs via finitely generated groups. Here, sofic groups and amenable groups are in the center of our considerations. Note that every finitely generated amenable group is sofic. We investigate the spectrum of a discrete self-adjoint operator by studying a sequence of finite dimensional analogues of these operators. In the setting of amenable groups we obtain these approximating operators by restricting the operator in question to finite subsets Qn , n ∈ N. These finite dimensional operators are self-adjoint and therefore admit a well-defined normalized eigenvalue counting function. The limit of the normalized eigenvalue counting functions when |Qn | → ∞ (if it exists) is called the integrated density of states (IDS). It is a distribution function of a probability measure encoding the distribution of the spectrum of the operator in question on the real axis. In this thesis, we prove the existence of the IDS in various geometric settings and for different types of operators. The models we consider include deterministic as well as random situations. Depending on the specific setting, we prove existence of the IDS as a weak limit of distribution functions or even as a uniform limit. Moreover, in certain situations we are able to express the IDS via a semi-explicit formula using the trace of the spectral projection of the original operator. This is sometimes referred to as the validity of the Pastur-Shubin trace formula. In the most general geometric setting we study, the operators are defined on Cayley graphs of sofic groups. Here we prove weak convergence of the eigenvalue counting functions and verify the validity of the Pastur-Shubin trace formula for random and non-random operators . These results apply to operators which not necessarily bounded or of finite hopping range. The methods are based on resolvent techniques. This theory is established without having an ergodic theorem for sofic groups at hand. Note that ergodic theory is the usual tool used in the proof of convergence results of this type. Specifying to operators on amenable groups we are able to prove stronger results. In the discrete case, we show that the IDS exists uniformly for a certain class of finite hopping range operators. This is obtained by using a Banach space-valued ergodic theorem. We show that this applies to eigenvalue counting functions, which implies their convergence with respect to the Banach space norm, in this case the supremum norm. Thus, the heart of this theory is the verification of the Banach space-valued ergodic theorem. Proceeding in two steps we first prove this result for so-called ST-amenable groups. Then, using results from the theory of ε-quasi tilings, we prove a version of the Banach space-valued ergodic theorem which is valid for all amenable groups. Focusing on random operators on amenable groups, we prove uniform existence of the IDS without the assumption that the operator needs to be of finite hopping range or bounded. Moreover, we verify the Pastur-Shubin trace formula. Here we present different techniques. First we show uniform convergence of the normalized eigenvalue counting functions adapting the technique of the Banach space-valued ergodic theorem from the deterministic setting. In a second approach we use weak convergence of the eigenvalue counting functions and additionally obtain control over the convergence at the jumps of the IDS. These ingredients are applied to verify uniform existence of the IDS. In both situations we employ results from the theory of large deviations, in order to deal with long-range interactions.

Cette thèse porte sur l'étude des points entiers et rationnels de certaines courbes et variétés modulaires. Après une brève introduction décrivant les motivations et le cadre de ce genre d'études ainsi que les résultats principaux de la thèse, le manuscrit se divise en trois parties. Le premier chapitre s'intéresse aux Q-courbes, et aux morphismes Gal(Q/Q) -> PGL2(Fp) qu'on peut leur associer pour tout p premier. Nous montrons que sous de bonnes hypothèses, pour p assez grand par rapport au discriminant du corps de définition de la Q-courbe, ce morphisme est surjectif, ce qui résout un cas particulier du problème d'uniformité de Serre (toujours ouvert en général). Les outils principaux du chapitre sont la méthode de Mazur (basée ici sur des résultats d'Ellenberg), la méthode de Runge et des théorèmes d'isogénie, suivant la structure de preuve de Bilu et Parent. Le second chapitre consiste en des estimations analytiques de sommes pondérées de valeurs de fonctions L de formes modulaires, dans l'esprit de techniques développées par Duke et Ellenberg. La motivation de départ d'un tel résultat est l'application de la méthode de Mazur dans le premier chapitre. Le troisième chapitre est consacré à la recherche de généralisations de la méthode de Runge pour des variétés de dimension supérieure. Nous y redémontrons un résultat de Levin inspiré de cette méthode, avant d'en prouver une forme assouplie dite "de Runge tubulaire", plus largement applicable. Dans l'optique de recherche de points entiers de variétés modulaires, nous en donnons enfin un exemple d'utilisation à la réduction d'une surface abélienne en produit de courbes elliptiques. / This thesis concerns the study of integral and rational points on some modular curves and varieties. After a brief introduction which describes the motivation and the setting of this topic as well as the main results of this thesis, the manuscript follows a threefold development. The first chapter focuses on Q-curves, and on the morphisms Gal(Q/Q) -> PGL2(Fp) that we can build with a Q-curve for every prime p. We prove that, under good hypotheses, for p large enough with respect to the discriminant of the definition field of the Q-curve, such a morphism is surjective, which solves a particular case of Serre's uniformity problem (still open in general). The main tools of the chapter are Mazur's method (based here on results of Ellenberg), Runge's method, and isogeny theorems, following the strategy of Bilu and Parent. The second chapter covers analytic estimates of weighted sums of L-function values of modular forms, in the fashion of techniques designed by Duke and Ellenberg. The initial goal of such a result is the application of Mazur's method in the first chapter. The third chapter is devoted to the search for generalisations of Runge's method for higherdimensional varieties. Here we prove anew a result of Levin inspired by this method, before proving an enhanced version called "tubular Runge", more generally applicable. In the perspective of studying integral points of modular varieties, we finally give an example of application of this theorem to the reduction of an abelian surface in a product of elliptic curves.

The escape of trajectories is a ubiquitous phenomenon in open dynamical systems and stochastic processes. If escape occurs repetitively for a statistical ensemble of trajectories, the population of remaining trajectories often undergoes an exponential decay characterised by the so-called escape rate. Its inverse defines the lifetime of the decaying state, which represents an intrinsic property of the system. This paradigm is fundamental to nucleation theory and reaction-rate theory in chemistry, physics, and biology.<p><p>In many circumstances, escape is activated by the presence of noise, which may be of internal or external origin. This is the case for thermally activated escape over a potential energy barrier and, more generally, for noise-induced escape in continuous-time or discrete-time dynamics. <p><p>In the weak-noise limit, the escape rate is often observed to decrease exponentially with the inverse of the noise amplitude, a behaviour which is given by the van't Hoff-Arrhenius law of chemical kinetics. In particular, the two important quantities to determine in this case are the exponential dependence (the ``activation energy') and its prefactor.<p><p>The purpose of the present thesis is to develop an analytical method to determine these two quantities. We consider in particular one-dimensional continuous and discrete-time systems perturbed by Gaussian white noise and we focus on the escape from the basin of attraction of an attracting fixed point.<p><p>In both classes of systems, using path-integral methods, a formula is deduced for the noise-induced escape rate from the attracting fixed point across an unstable fixed point, which forms the boundary of the basin of attraction. The calculation starts from the trace formula for the eigenvalues of the operator ruling the time evolution of the probability density in noisy maps. The escape rate is determined by the loop formed by two heteroclinic orbits connecting back and forth the two fixed points in a two-dimensional auxiliary deterministic dynamical system. The escape rate is obtained, including the expression of the prefactor to van't Hoff-Arrhenius exponential factor./L'échappement des trajectoires est un phénomène omniprésent dans les systèmes dynamiques ouverts et les processus stochastiques. Si l'échappement se produit de façon répétitive pour un ensemble statistique de trajectoires, la population des trajectoires restantes subit souvent une décroissance exponentielle caractérisée par le taux d'échappement. L'inverse du taux d'échappement définit alors la durée de vie de l'état transitoire associé, ce qui représente une propriété intrinsèque du système. Ce paradigme est fondamental pour la théorie de la nucléation et, de manière générale, pour la théorie des taux de transitions en chimie, en physique et en biologie.<p><p>Dans de nombreux cas, l'échappement est induit par la présence de bruit, qui peut être d'origine interne ou externe. Ceci concerne en particulier l'échappement activé thermiquement à travers une barrière d'énergie potentielle, et plus généralement, l'échappement dû au bruit dans les systèmes dynamiques à temps continu ou à temps discret.<p><p>Dans la limite de faible bruit, on observe souvent une décroissance exponentielle du taux d'échappement en fonction de l'inverse de l'amplitude du bruit, un comportement qui est régi par la loi de van't Hoff-Arrhenius de la cinétique chimique. En particulier, les deux quantités importantes de cette loi sont le coefficient de la dépendance exponentielle (c'est-à-dire ``l'énergie d'activation') et son préfacteur.<p><p>L'objectif de cette thèse est de développer une théorie analytique pour déterminer ces deux quantités. La théorie que nous présentons concerne les systèmes unidimensionnels à temps continu ou discret perturbés par un bruit blanc gaussien et nous considérons le problème de l'échappement du bassin d'attraction d'un point fixe attractif. Pour s'échapper, les trajectoires du système bruité initialement contenues dans ce bassin d'attraction doivent alors traverser un point fixe instable qui forme la limite du bassin.<p><p>Dans le présent travail, et pour les deux types de systèmes, une formule est dérivée pour le taux d'échappement du point fixe attractif en utilisant des méthodes d'intégrales de chemin. Le calcul utilise la formule de trace pour les valeurs propres de l'opérateur gouvernant l'évolution temporelle de la densité de probabilité dans le système bruité. Le taux d'échappement est déterminé en considérant la boucle formée par deux orbites hétéroclines liant dans les deux sens les deux points fixes dans un système dynamique auxiliaire symplectique et bidimensionnel. On obtient alors le taux d'échappement, comprenant l'expression du préfacteur de l'exponentielle de la loi de van't Hoff-Arrhenius. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

This thesis is devoted to the study of operators on discrete structures. The operators are supposed to be self-adjoint and obey a certain translation invariance property. The discrete structures are given as Cayley graphs via finitely generated groups. Here, sofic groups and amenable groups are in the center of our considerations. Note that every finitely generated amenable group is sofic. We investigate the spectrum of a discrete self-adjoint operator by studying a sequence of finite dimensional analogues of these operators. In the setting of amenable groups we obtain these approximating operators by restricting the operator in question to finite subsets Qn , n ∈ N. These finite dimensional operators are self-adjoint and therefore admit a well-defined normalized eigenvalue counting function. The limit of the normalized eigenvalue counting functions when |Qn | → ∞ (if it exists) is called the integrated density of states (IDS). It is a distribution function of a probability measure encoding the distribution of the spectrum of the operator in question on the real axis. In this thesis, we prove the existence of the IDS in various geometric settings and for different types of operators. The models we consider include deterministic as well as random situations. Depending on the specific setting, we prove existence of the IDS as a weak limit of distribution functions or even as a uniform limit. Moreover, in certain situations we are able to express the IDS via a semi-explicit formula using the trace of the spectral projection of the original operator. This is sometimes referred to as the validity of the Pastur-Shubin trace formula. In the most general geometric setting we study, the operators are defined on Cayley graphs of sofic groups. Here we prove weak convergence of the eigenvalue counting functions and verify the validity of the Pastur-Shubin trace formula for random and non-random operators . These results apply to operators which not necessarily bounded or of finite hopping range. The methods are based on resolvent techniques. This theory is established without having an ergodic theorem for sofic groups at hand. Note that ergodic theory is the usual tool used in the proof of convergence results of this type. Specifying to operators on amenable groups we are able to prove stronger results. In the discrete case, we show that the IDS exists uniformly for a certain class of finite hopping range operators. This is obtained by using a Banach space-valued ergodic theorem. We show that this applies to eigenvalue counting functions, which implies their convergence with respect to the Banach space norm, in this case the supremum norm. Thus, the heart of this theory is the verification of the Banach space-valued ergodic theorem. Proceeding in two steps we first prove this result for so-called ST-amenable groups. Then, using results from the theory of ε-quasi tilings, we prove a version of the Banach space-valued ergodic theorem which is valid for all amenable groups. Focusing on random operators on amenable groups, we prove uniform existence of the IDS without the assumption that the operator needs to be of finite hopping range or bounded. Moreover, we verify the Pastur-Shubin trace formula. Here we present different techniques. First we show uniform convergence of the normalized eigenvalue counting functions adapting the technique of the Banach space-valued ergodic theorem from the deterministic setting. In a second approach we use weak convergence of the eigenvalue counting functions and additionally obtain control over the convergence at the jumps of the IDS. These ingredients are applied to verify uniform existence of the IDS. In both situations we employ results from the theory of large deviations, in order to deal with long-range interactions.