Let K be a number field with ring of integers O_K and let G be a finite group.By a result of E. Noether, the ring of integers of a tame Galois extension of K with Galois group G is a locally free O_K[G]-module of rank 1.Thus, to any tame Galois extension L/K with Galois group G we can associate a class [O_L] in the locally free class group Cl(O_K[G]). The set of all classes in Cl(O_K[G]) which can be obtained in this way is called the set of realizable classes and is denoted by R(O_K[G]).In this dissertation we study different problems related to R(O_K[G]).The first part focuses on the following question: is R(O_K[G]) a subgroup of Cl(O_K[G])? When the group G is abelian, L. McCulloh proved that R(O_K[G]) coincides with the so-called Stickelberger subgroup St(O_K[G]) of Cl(O_K[G]). In Chapter 2, we give a detailed presentation of unpublished work by L. McCulloh that extends the definition of St(O_K[G]) to the non-abelian case and shows that R(O_K[G]) is contained in St(O_K[G]) (the opposite inclusion is still not known in the non-abelian case).Then, just using its definition and Stickelberger's classical theorem, we prove in Chapter 3 that St(O_K[G]) is trivial if K=Q and G is either cyclic of order p or dihedral of order 2p, where p is an odd prime number. This, together with McCulloh's results, allows us to have a new proof of the triviality of R(O_K[G]) in the cases just considered.The main original results are contained in the second part of this thesis. In Chapter 4, we prove that St(O_K[G]) has good functorial behavior under restriction of the base field. This has the interesting consequence that, if N/L is a tame Galois extension with Galois group G, and St(O_K[G]) is known to be trivial for some subfield K of L, then O_N is stably free as an O_K[G]-module.In the last chapter, we prove an equidistribution result for Galois module classes amongst tame Galois extensions of K with Galois group G in which a given prime p of K is totally split.

This thesis extends a localisation theory for finite algebras to certain classes of infinite structures. Based on ideas and constructions originally stemming from Tame Congruence Theory, algebras are studied via local restrictions of their relational counterpart (Relational Structure Theory). In this respect, first those subsets are identified that are suitable for such a localisation process, i. e. that are compatible with the relational clone structure of the counterpart of an algebra. It is then studied which properties of the global algebra can be transferred to its localisations, called neighbourhoods. Thereafter, it is discussed how this process can be reversed, leading to the concept of covers. These are collections of neighbourhoods that allow information retrieval about the global structure from knowledge about the local restrictions. Subsequently, covers are characterised in terms of a decomposition equation, and connections to categorical equivalences of algebras are explored. In the second half of the thesis, a refinement concept for covers is introduced in order to find optimal, non-refinable covers, eventually leading to practical algorithms for their determination. Finally, the text establishes further theoretical foundations, e. g. several irreducibility notions, in order to ensure existence of non-refinable covers via an intrinsic characterisation, and to prove under some conditions that they are uniquely determined in a canonical sense. At last, the applicability of the developed techniques is demonstrated using two clear expository examples. / Diese Dissertation erweitert eine Lokalisierungstheorie für endliche Algebren auf gewisse Klassen unendlicher Strukturen. Basierend auf Ideen und Konstruktionen, die ursprünglich der Tame Congruence Theory entstammen, werden Algebren über lokale Einschränkungen ihres relationalen Gegenstücks untersucht (Relationale Strukturtheorie). In diesem Zusammenhang werden zunächst diejenigen Teilmengen identifiziert, welche für einen solchen Lokalisierungsprozeß geeignet sind, d. h., die mit der Relationenklonstruktur auf dem Gegenstück einer Algebra kompatibel sind. Es wird dann untersucht, welche Eigenschaften der globalen Algebra auf ihre Lokalisierungen, genannt Umgebungen, übertragen werden können. Nachfolgend wird diskutiert, wie dieser Vorgang umgekehrt werden kann, was zum Begriff der Überdeckungen führt. Dies sind Systeme von Umgebungen, welche die Rückgewinnung von Informationen über die globale Struktur aus Kenntnis ihrer lokalen Einschränkungen erlauben. Sodann werden Überdeckungen durch eine Zerlegungsgleichung charakterisiert und Bezüge zu kategoriellen Äquivalenzen von Algebren hergestellt. In der zweiten Hälfte der Arbeit wird ein Verfeinerungsbegriff für Überdeckungen eingeführt, um optimale, nichtverfeinerbare Überdeckungen zu finden, was letztlich zu praktischen Algorithmen zu ihrer Bestimmung führt. Schließlich erarbeitet der Text weitere theoretische Grundlagen, beispielsweise mehrere Irreduzibilitätsbegriffe, um die Existenz nichtverfeinerbarer Überdeckungen vermöge einer intrinsischen Charakterisierung sicherzustellen und, unter gewissen Bedingungen, zu beweisen, daß sie in kanonischer Weise eindeutig bestimmt sind. Schlußendlich wird die Anwendbarkeit der entwickelten Methoden an zwei übersichtlichen Beispielen demonstriert.

L’objet de cette thèse est de comprendre comment se généralise la théorie de la ramification pour des actions par des schémas en groupes affines avec un intérêt particulier pour la notion de modération. Comme contexte général pour ce résumé, considérons une base affine S := Spec(R) où R est un anneau unitaire, commutatif, X := Spec(B) un schéma affine sur S, G := Spec(A) un schéma en groupes affine, plat et de présentation finie sur S et une action de G sur X que nous noterons (X, G). Enfin, nous notons [X/G] le champ quotient associé à cette action et Y := Spec(BA) où BA est l’anneau des invariants pour l’action (X, G). Supposons de plus que le champ d’inertie soit fini.Comme point de référence, nous prenons la théorie classique de la ramification pour des anneaux munis d’une action par un groupe fini abstrait. Afin de comprendre comment généraliser cette théorie pour des actions par des schémas en groupes, nous considérons les actions par des schémas en groupes constants en se rappelant que la donnée de telles actions est équivalente à celle d’un anneau muni d’une action par un groupe fini abstrait nous ramenant au cas classique. Nous obtenons ainsi dans ce nouveau contexte des notions généralisant l’anneau des invariants en tant que quotient, les groupes d’inertie et toutes leurs propriétés. Le cas non ramifié se généralise naturellement avec les actions libres. En ce qui concerne le cas modéré, qui nous intéresse particulièrement pour cette thèse, deux généralisations sont proposées dans la littérature. Celle d’actions modérées par des schémas en groupes affines introduite par Chinburg, Erez, Pappas et Taylor dans l’article [CEPT96] et celle de champ modéré introduite par Abramovich, Olsson et Vistoli dans [AOV08]. Il a été alors naturel d’essayer de comparer ces deux notions et de comprendre comment se généralisent les propriétés classiques d’objets modérés à des actions par des schémas en groupes affines.Tout d’abord, nous avons traduit algébriquement la propriété de modération sur un champ quotient comme l’exactitude du foncteur des invariants. Ce qui nous a permis d’obtenir aisément à l’aide de [CEPT96] qu’une action modérée définit toujours un champ quotient modéré. Quant à la réciproque, nous avons réussi à l’obtenir seulement lorsque nous supposons de plus que G est fini et localement libre sur S et que X est plat sur Y . Nous pouvons voir que la notion de modération pour l’anneau B muni d’une action par un groupe fini abstrait Γ est équivalente au fait que tous les groupes d’inertie aux points topologiques sont linéairement réductifs si l’on considère l’action par le schéma en groupes constant correspondant à Γ sur X. Il a été donc naturel de se demander si cette propriété est encore vraie en général. Effectivement, l’article [AOV08] caractérise le fait que le champ quotient [X/G] est modéré par le fait que les groupes d’inertie aux points géométriques sont linéairement réductifs.À nouveau, si l’on considère le cas des anneaux munis d’une action par un groupe fini abstrait, il est bien connu que l’action peut être totalement reconstruite à partir de l’action d’un groupe inertie. Lorsque l’on considère le cas des actions par les schémas en groupes constants, cela se traduit comme un théorème de slices, c’est-à-dire une description locale de l’action initiale par une action par un groupe d’inertie. Par exemple, lorsque G est fini, localement libre sur S, nous établissons que le fait qu’une action soit libre est une propriété locale pour la topologie fppf, ce qui peut se traduire comme un théorème de slices. Grâce à [AOV08], nous savons déjà qu’un champ quotient modéré [X/G] est localement isomorphe pour la topologie fppf à un champ quotient [X/H] où H est une extension du groupe d’inertie en un point de Y. Lorsque G est fini sur S, il nous a été possible de montrer que H est aussi un sous-groupe de G. / The purpose of this thesis is to understand how to generalize the ramification theory for actions by affine group schemes with a particular interest for the notion of tameness. As general context for this summary, we consider an affine basis S := Spec(R) where R is a commutative, unitary ring, an affine, finitely presented, Noetherian scheme X := Spec(B) over S, a flat, finitely presented, affine group scheme G := Spec(A) over S and an action of G on X that we denote by (X, G). Finally, we denote [X/G] the quotient stack associated to this action and we set Y := Spec(BA) where BA is the ring of invariants for the action (X, G). Moreover, we suppose that the inertia stack is finite.As reference point, we take the classical theory of ramification for rings endowed with an action of a finite, abstract group. In order to understand how to generalize this theory for actions of group schemes, we consider the actions of constant group schemes knowing that the data of such actions is equivalent to the data of rings endowed with an action of a finite abstract group, this being the classical case. We obtain thus in this new context notions generalizing the ring of invariants as a quotient, the inertia group and all their properties. The unramified case is generalized naturally by the free actions. For the tame case, which interests us particularly here, two generalizations are proposed in the literature: the one of tame actions of affine group schemes introduced by Chinburg, Erez, Pappas et Taylor in the article [CEPT96] and the one of tame stacks introduced by Abramovich, Olsson and Vistoli in [AOV08]. It was then natural to compare these two notions and to understand how to generalize the classical properties of tame objects for the actions of affine group schemes. First of all, we traduced algebraically the tameness property on a quotient stack as the exactness of the functor of invariants. This permits to obtain easily thanks to [CEPT96] that tame actions define always tame quotient stacks. For the converse, we only manage to prove it when we suppose G to be finite, locally free over S and X flat over Y . We are able to see that the notion of tameness for a ring endowed with an action of a finite, abstract group Γ is equivalent to the fact that all the inertia group schemes at the topological points are linearly reductive if we consider the action of the constant group scheme corresponding to Γ over X. It was thus natural to wonder if this property was also true in general. In fact, the article [AOV08] characterizes the fact that the quotient stack [X/G] is tame by the fact that the inertia group schemes at the geometric points are linearly reductive.Again, if we consider the case of rings endowed with an action of a finite, abstract group, it is well known that these actions can be totally reconstructed from an action involving an inertia group. When we consider actions by constant group schemes, this is translated as a slice theorem, that is, a local description of the initial action by an action involving an inertia group. For example, we establish that the fact that an action is free is a "local property" for the fppf topology and this can be translated also as a "local" slice theorem. Thanks to [AOV08], we already know that a tame quotient stack [X/G] is locally isomorphic for the fppf topology to a quotient stack [X/H], where H is an extension of the inertia group in a point of Y . When G is finite over S, it was possible to show that H is also a subgroup of G. In this thesis, it was not possible to obtain a slice theorem in this generality. However, when G is commutative, finite over S, it is possible to prove the existence of a torsor, if we suppose [X/G] to be tame. This permits to prove a slice theorem when G is commutative, finite over S and [X/G] is tame. / Lo scopo di questa tesi è capire come si generalizza la teoria della ramificazione per azioni di schemi in gruppi affini con un interesse particolare per la nozione di moderazione. Come contesto generale per questo riassunto, consideriamo una base affine S := Spec(R) dove R è un anello unitario e commutativo, X := Spec(B) uno schema affine, noetheriano e di presentazione finita su S, G := Spec(A) uno schema in gruppi affine, piatto e di presentazione finita su S e un’azione di G su X che denoteremo (X, G). Infine, denotiamo con [X/G] lo stack quoziente associato a questa azione e Y := Spec(BA) dove BA è l’anello degli invarianti per l’azione (X, G). Supponiamo inoltre che il campo d’inerzia sia finito.Come punto di riferimento prendiamo la teoria classica della ramificazione per anelli muniti d’un’azione d’un gruppo finito astratto. Al fine di comprendere come generalizzare questa teoria per azioni di schemi in gruppi, consideriamo le azioni di schemi in gruppi costanti ricordando che il dato di tali azioni è equivalente al dato d’un anello dotato d’un’azione d’un gruppo finito astratto, riconducendosi al caso classico. Otteniamo così in questo nuovo contesto delle nozioni che generalizzano l’anello degli invarianti in quanto quoziente, i gruppi d’inerzia e tutte le loro proprietà. Il caso non ramificato si generalizza in modo naturale con le azioni libere. Per qual che riguarda il caso moderato, al quale siamo particolarmente interessati in questa tesi, due generalizzazioni sono proposte nella letteratura: quella delle azioni moderate di schemi in gruppi affini introdotta da Chinburg, Erez, Pappas e Taylor nell’articolo [CEPT96] e quella di stack moderato introdotta da Abramovich, Olsson e Vistoli in [AOV08]. È stato quindi naturale cercare di confrontare queste due nozioni e capire come si generalizzano le proprietà classiche degli oggetti moderati ad azioni di schemi in gruppi affini.Per cominciare, abbiamo tradotto algebricamente la proprietà di moderazione su un stack quoziente come l’esattezza del funtore degli invarianti. Ciò ha permesso d’ottenere agevolmente, usando [CEPT96], che un’azione moderata definisce sempre uno stack quoziente moderato. Quanto al viceversa, siamo riusciti ad ottenerlo solamente sotto l’ulteriore ipotesi che G sia finito e localmente libero su S e che X sia piatto su Y . Possiamo vedere che la nozione di moderazione per l’anello B dotato d’un’azione d’un gruppo finito astratto Γ è equivalente al fatto che tutti i gruppi d’inerzia sui punti topologici siano linearmente riduttivi se si considera l’azione dello schema in gruppi costante corrispondente a Γ su X. È stato quindi naturale domandarsi se questa proprietà sia vera in generale. In effetti, l’articolo [AOV08] caratterizza il fatto che lo stack quoziente [X/G] è moderato tramite il fatto che i gruppi d’inerzia sui punti geometrici siano linearmente riduttivi.Di nuovo, se consideriamo il caso degli anelli muniti d’un’azione d’un gruppo finito astratto, è ben noto che quest’azione può essere totalmente ricostruita a partire da un’azione in cui interviene un gruppo d’inerzia. Quando consideriamo il caso delle azioni degli schemi in gruppi costanti, questo si traduce come un teorema di slices, cioè una descrizione locale dell’azione di partenza (X,G) tramite un’azione in cui interviene un gruppo d’inerzia. Per esempio quando G è finito e localmente libero su S, stabiliamo che il fatto che un’azione è libera è una proprietà locale per la topologia fppf, ciò si può interpretare come un teorema di slices. Grazie a [AOV08] sappiamo già che uno stack quoziente moderato [X/G] è localmente isomorfo per la topologia fppf a uno stack quoziente [X/H], dove H è un’estensione d’un gruppo d’inerzia in un punto di Y.

This thesis extends a localisation theory for finite algebras to certain classes of infinite structures. Based on ideas and constructions originally stemming from Tame Congruence Theory, algebras are studied via local restrictions of their relational counterpart (Relational Structure Theory). In this respect, first those subsets are identified that are suitable for such a localisation process, i. e. that are compatible with the relational clone structure of the counterpart of an algebra. It is then studied which properties of the global algebra can be transferred to its localisations, called neighbourhoods. Thereafter, it is discussed how this process can be reversed, leading to the concept of covers. These are collections of neighbourhoods that allow information retrieval about the global structure from knowledge about the local restrictions. Subsequently, covers are characterised in terms of a decomposition equation, and connections to categorical equivalences of algebras are explored. In the second half of the thesis, a refinement concept for covers is introduced in order to find optimal, non-refinable covers, eventually leading to practical algorithms for their determination. Finally, the text establishes further theoretical foundations, e. g. several irreducibility notions, in order to ensure existence of non-refinable covers via an intrinsic characterisation, and to prove under some conditions that they are uniquely determined in a canonical sense. At last, the applicability of the developed techniques is demonstrated using two clear expository examples.:1 Introduction 2 Preliminaries and Notation 2.1 Functions, operations and relations 2.2 Algebras and relational structures 2.3 Clones 3 Relational Structure Theory 3.1 Finding suitable subsets for localisation 3.2 Neighbourhoods 3.3 The restricted algebra A|U 3.4 Covers 3.5 Refinement 3.6 Irreducibility notions 3.7 Intrinsic description of non-refinable covers 3.8 Elaborated example 4 Problems and Prospects for Future Research Acknowledgements Index of Notation Index of Terms Bibliography / Diese Dissertation erweitert eine Lokalisierungstheorie für endliche Algebren auf gewisse Klassen unendlicher Strukturen. Basierend auf Ideen und Konstruktionen, die ursprünglich der Tame Congruence Theory entstammen, werden Algebren über lokale Einschränkungen ihres relationalen Gegenstücks untersucht (Relationale Strukturtheorie). In diesem Zusammenhang werden zunächst diejenigen Teilmengen identifiziert, welche für einen solchen Lokalisierungsprozeß geeignet sind, d. h., die mit der Relationenklonstruktur auf dem Gegenstück einer Algebra kompatibel sind. Es wird dann untersucht, welche Eigenschaften der globalen Algebra auf ihre Lokalisierungen, genannt Umgebungen, übertragen werden können. Nachfolgend wird diskutiert, wie dieser Vorgang umgekehrt werden kann, was zum Begriff der Überdeckungen führt. Dies sind Systeme von Umgebungen, welche die Rückgewinnung von Informationen über die globale Struktur aus Kenntnis ihrer lokalen Einschränkungen erlauben. Sodann werden Überdeckungen durch eine Zerlegungsgleichung charakterisiert und Bezüge zu kategoriellen Äquivalenzen von Algebren hergestellt. In der zweiten Hälfte der Arbeit wird ein Verfeinerungsbegriff für Überdeckungen eingeführt, um optimale, nichtverfeinerbare Überdeckungen zu finden, was letztlich zu praktischen Algorithmen zu ihrer Bestimmung führt. Schließlich erarbeitet der Text weitere theoretische Grundlagen, beispielsweise mehrere Irreduzibilitätsbegriffe, um die Existenz nichtverfeinerbarer Überdeckungen vermöge einer intrinsischen Charakterisierung sicherzustellen und, unter gewissen Bedingungen, zu beweisen, daß sie in kanonischer Weise eindeutig bestimmt sind. Schlußendlich wird die Anwendbarkeit der entwickelten Methoden an zwei übersichtlichen Beispielen demonstriert.:1 Introduction 2 Preliminaries and Notation 2.1 Functions, operations and relations 2.2 Algebras and relational structures 2.3 Clones 3 Relational Structure Theory 3.1 Finding suitable subsets for localisation 3.2 Neighbourhoods 3.3 The restricted algebra A|U 3.4 Covers 3.5 Refinement 3.6 Irreducibility notions 3.7 Intrinsic description of non-refinable covers 3.8 Elaborated example 4 Problems and Prospects for Future Research Acknowledgements Index of Notation Index of Terms Bibliography

Soit K/F une extension quadratique modérément ramifiée de corps locaux non archimédiens. Soit GLm (D) une forme intérieure de GLn (F) et GLμ (∆) = (Mm (D) ⊗ K)× . Alors GLμ (∆) est une forme intérieure de GLn (K), les quotients GLμ (∆)/GLm (D) et GLn (K)/GLn (F) sont des espaces symétriques. En utilisant la paramétrisation de Silberger et Zink, nous déterminons des critères de GLm (D)-distinction pour les cuspidales de niveau 0 de GLμ (∆), puis nous prouvons qu’une cuspidale de niveau 0 de GLn (K) est GLn (F)-distinguée si et seulement si son image par la correspondance de Jacquet-Langlands est GLm (D)-distinguée. Puis, dans le cas particulier où μ = 2 et m = 1, nous regardons le cas des séries discrètes de niveau 0 non cuspidales, en utilisant le système de coefficients sur l’immeuble associé à la représentation, donné par Schneider et Stuhler. / Let K/F be a tamely ramified quadratic extension of non-archimedean locally compact fields. Let GLm (D) be an inner form of GLn (F) and GLμ (∆) = (Mm (D)⊗K)× . Then GLμ (∆) is an inner form of GLn (K), the quotients GLμ (∆)/GLm (D) and GLn (K)/GLn (F) are symmetric spaces. Using the parametrization of Silberger and Zink, we determine conditions of GLm (D)-distinction for level zero cuspidal representations of GLμ (∆). We also show that a level zero cuspidal representation of GLn (K) is GLn (F)-distinguished if and only if its image by the Jacquet-Langlands correspondence is GLm (D)-distinguished. Then, we treat the case of level zero non supercuspidal representations when μ = 2 and m = 1 using the coefficient system of the Bruhat-Tits building associated to the representation by Schneider and Stuhler.

Freeways have had a strong influence not only on the urban transportation but also on downtown areas both physically and socially. Certainly, they have extended the commuting limits of the city and made lower land costs more accessible. However, many of the mid-century freeways, once championed by planners as tools for urban renewal, have created swaths of blight through city neighborhoods. Their negative impacts on the larger urban framework requires new ideas for healthier alternatives to aid in preserving and building sustainable cities. Removal of any downtown highway requires careful thought— even more consideration than when it was built. Quick solutions are what resulted in the problems that downtown highways of the Interstate-Era have today. If it is the simple interactions between people and place are that make up the positive aspects an urban environment, then what are the possibilities and strategies for removing urban highway, which are one of the primary impediments separating people in place in contemporary cities? This question is the focus of this thesis. At its core, the removal of freeways represents a trade-off between mobility objectives and economic development objectives. Evidence from other cities’ decisions to redesign or remove their downtown highways suggests multiple benefits. Making design changes, such as to replace a downtown highway with a well-designed surface boulevard, can stimulate economic activities without necessarily causing traffic chaos. Solutions come in different shapes and sizes. The selected case studies in this thesis reflect a diversity of approaches – suggesting no single strategy exists for addressing downtown highway issues. This reflects the fact that multiple alternatives must be considered in every situation because each approach varies in costs and opportunities. A typology of highway alternations derived from the case studies includes seven different techniques: burying, demolishing, taming, capping or bridging, elevating, retaining, and relocating. The final chapter applies the conclusions from the case studies to the Downtown Connector– Interstate 75/85– in Downtown Atlanta, Georgia. Urban design and transportation planning has an emerging new set of values. Transportation planning is seeking to promote alternate modes of transportation to the private vehicle, like transit, by foot, or by bicycle. We now understand that connectivity is not served only by highways but also by urban street networks that invite modes other than just automobiles. An important role for urban design will be to shape the way these interactions are made to benefit the citizens, its urban spaces, and the economy.

L’objet de cette thèse est l’étude de la propagation de la 2-birationalité pour les 2-extensions de corps de nombres. Le problème étudié se présente comme suit : étant donnés un corps 2-rationnel totalement réel K, une extension quadratique totalement imaginaire L de K, et une 2-extension totalement réelle de K de K, à quelles conditions la 2-birationalité du compositum L = KL se lit-elle sur L ? La thèse se structure en trois parties : l’étude du cas absolument quadratique d’abord, le cas relativement quadratique ensuite ; le cas général enfin. Le résultat principal de la thèse résout complètement le problème posé en toute généralité. En fin de thèse, diverses illustrations numériques sont proposées à l’aide du PARI, ainsi qu’une étude des tours d’extensions 2-birationnelles. / This thesis deals with the propagation of 2-birationality for 2-extensions of numbers fields. More precisely, le t K be a 2-rational totally real number field, L a CM quadratic extension of K, and let K be a totally real 2-extension of K. Under which conditions can one read the 2-birationaltiy of the compositum L = LK from L ? This work is divided into three parts : we first study the absolute quadratic case, then the relatively quadratic case, then finally the general case. The thesis’s main result solves the whole problem. We also illustrate the result with various numeric examples, obtained with PARI and a focus at the end on 2-birational extensions’ towers.

Soient K un corps de nombres d'anneau des entiers O_K et G un groupe fini. Grâce à un résultat de E. Noether, l'anneau des entiers d'une extension galoisienne de K modérément ramifiée, de groupe de Galois G, est un O_K[G]-module localement libre de rang 1. Donc, à chaque extension galoisienne L/K modérément ramifiée, de groupe de Galois G, on peut associer une classe [O_L] dans le groupe des classes des modules localement libres Cl(O_K[G]). L'ensemble des classes de Cl(O_K[G]) qui peuvent être obtenues de cette façon est appelé ensemble des classes réalisables et on le note R(O_K[G]).Dans cette thèse, on étudie différents problèmes liés à R(O_K[G]). Dans la première partie, nous nous focalisons sur la question suivante: R(O_K[G]) est-il un sous-groupe de Cl(O_K[G])? Si G est abélien, L. McCulloh a prouvé que R(O_K[G]) coïncide avec le soi-disant sous-groupe de Stickelberger St(O_K[G]) dans Cl(O_K[G]). Dans le Chapitre 2, nous donnons une présentation détaillée d'un travail non publié de L. McCulloh qui étend la définition de St(O_K[G]) au cas non-abélien et montre que R(O_K[G]) est inclus dans St(O_K[G]) (l'inclusion opposée n'est pas encore connue dans le cas non-abélien). Puis, en utilisant sa définition et le Théorème de Stickelberger classique, nous montrons dans le Chapitre 3 que St(O_K[G]) est trivial si K=Q et G est soit un groupe cyclique d'ordre p soit un groupe diédral d'ordre 2p, avec p premier impair. Ceci, lié aux résultats de McCulloh, nous donne une nouvelle preuve de la trivialité de R(O_K[G]) dans les cas considérés.Les résultats originaux les plus importants sont contenus dans la deuxième partie de cette thèse. Dans le Chapitre 4 nous montrons la fonctorialité de St(O_K[G]) par rapport au changement du corps de base. Ceci implique que si N/L est une extension galoisienne modérément ramifiée, de groupe de Galois G, et St(O_K[G]) est connu être trivial pour un certain sous-corps K de L, alors O_N est un O_K[G]-module stablement libre.Dans le dernier chapitre, nous montrons un résultat concernant la distribution des classes réalisables parmi les extensions galoisiennes de K modérément ramifiées, de groupe de Galois G, dans lesquelles un idéal premier de K donné est totalement décomposé. / Let K be a number field with ring of integers O_K and let G be a finite group.By a result of E. Noether, the ring of integers of a tame Galois extension of K with Galois group G is a locally free O_K[G]-module of rank 1.Thus, to any tame Galois extension L/K with Galois group G we can associate a class [O_L] in the locally free class group Cl(O_K[G]). The set of all classes in Cl(O_K[G]) which can be obtained in this way is called the set of realizable classes and is denoted by R(O_K[G]).In this dissertation we study different problems related to R(O_K[G]).The first part focuses on the following question: is R(O_K[G]) a subgroup of Cl(O_K[G])? When the group G is abelian, L. McCulloh proved that R(O_K[G]) coincides with the so-called Stickelberger subgroup St(O_K[G]) of Cl(O_K[G]). In Chapter 2, we give a detailed presentation of unpublished work by L. McCulloh that extends the definition of St(O_K[G]) to the non-abelian case and shows that R(O_K[G]) is contained in St(O_K[G]) (the opposite inclusion is still not known in the non-abelian case).Then, just using its definition and Stickelberger's classical theorem, we prove in Chapter 3 that St(O_K[G]) is trivial if K=Q and G is either cyclic of order p or dihedral of order 2p, where p is an odd prime number. This, together with McCulloh's results, allows us to have a new proof of the triviality of R(O_K[G]) in the cases just considered.The main original results are contained in the second part of this thesis. In Chapter 4, we prove that St(O_K[G]) has good functorial behavior under restriction of the base field. This has the interesting consequence that, if N/L is a tame Galois extension with Galois group G, and St(O_K[G]) is known to be trivial for some subfield K of L, then O_N is stably free as an O_K[G]-module.In the last chapter, we prove an equidistribution result for Galois module classes amongst tame Galois extensions of K with Galois group G in which a given prime p of K is totally split.

We study a multivariate version of the Bernstein–Sato polynomial, the so-called Bernstein–Sato ideal, associated to an arbitrary factorization of an analytic germ <i>f - f</i>₁···<i>f</i>_r. We identify a large class of geometrically characterized germs so that the <i>D</i>_X,x[<i>s</i>₁,...,<i>s</i>_r]-annihilator of <i>f</i>^s₁¹···<i>f</i>^s_r^r admits the simplest possible description and, more-over, has a particularly nice associated graded object. As a consequence we are able to verify Budur’s Topological Multivariable Strong Monodromy Conjecture for arbitrary factorizations of tame hyperplane arrangements by showing the zero locus of the associated Bernstein–Sato ideal contains a special hyperplane. By developing ideas of Maisonobe and Narvaez-Macarro, we are able to find many more hyperplanes contained in the zero locus of this Bernstein–Sato ideal. As an example, for reduced, tame hyperplane arrangements we prove the roots of the Bernstein–Sato polynomial contained in [−1,0) are combinatorially determined; for reduced, free hyperplane arrangements we prove the roots of the Bernstein–Sato polynomial are all combinatorially determined. Finally, outside the hyperplane arrangement setting, we prove many results about a certain <i>D</i>_X,x-map ∇_<i>A</i> that is expected to characterize the roots of the Bernstein–Sato ideal.

Cette thèse comporte trois chapitres indépendants portant sur l’itération des applicationsrationnelles sur des variétés projectives et plus spécifiquement sur l’étude du comportement dela suite des degrés des itérés de telles applications.Dans le premier chapitre, nous donnons une construction des invariants fondamentaux quesont les degrés dynamiques dans un cadre très général, et ce sans hypothèse ni sur la caractéristique ni sur les singularités de l’espace ambiant. Cette construction repose sur des propriétésde positivité des cycles algébriques, et propose une alternative aux approches analytiques deDinh et Sibony ou algébriques de Truong.Le second chapitre est issu d’un article écrit en commun avec Jian Xiao. Notre contributionporte sur des objets centraux en géométrie convexe appelés valuations. Nous transférons à l’espace des valuations des notions de positivité des cycles algébriques récemment introduites parLehmann et Xiao, ce qui nous permet d’étendre l’opération de convolution originellement définie par Bernig et Fu à une sous-classe de valuations suffisamment positives.Le troisième chapitre constitue le coeur de la thèse, et porte sur des estimations des degrésdynamiques des automorphismes dit modérés de la quadrique affine de dimension 3. Nos arguments sont de nature variée, et s’appuient sur l’action du groupe modéré sur un complexe carréCAT(0) et Gromov hyperbolique récemment introduite par Bisi, Furter et Lamy.Nous avons finalement collecté dans un dernier et court chapitre quelques pistes de recherchedirectement inspirées des travaux présentés ici. / This thesis is divided into three independent chapters on the iterates of rational maps on projective varieties and more specifically on the study of the growth of the degree sequences of the iterates of such maps. In the first chapter, we give a construction of the fundamental invariants called dynamical degrees. Our method holds in a very general setting, without any conditions on the characteristic of the field or on the singularities of the ambient space.This construction is based on the study of positivity properties of algebraic cycles and gives an alternative approach to the analytical technics of Dinh and Sibony or to the algebraic arguments of Truong.The second chapter is taken from an article written in joint work with Jian Xiao. Our paper focuses on central objects in convex geometry called valuations. We transfer some positivity notions of algebraic cycles recently introduced by Lehmann and Xiao, this allows us to extend the convolution operation defined by Bernig and Fu to a subspace of sufficiently positive valuations.The third chapter is the core of this thesis and focuses on the dynamical degrees of the so-called tame automorphisms of an affine quadric threefold. Our arguments are of various nature and rely on the action of the tame group on a CAT(0), Gromov hyperbolic square complex recently introduced by Bisi, Furter and Lamy. Finally, we have collected in the last chapter a few perpectives directly inspired by this work.