Uniformity and Functional Equations for Local Zeta Functions of K\mathfrak{K}-Split Algebraic Groups

We study the local zeta functions of an algebraic group $\mathcal{G}$ defined over $\mathfrak{K}$ together with a faithful $\mathfrak{K}$-rational representation $\rho$ for a finite extension $\mathfrak{K}$ of $\mathbb{Q}$. These are given by integrals over $\mathfrak{p}$-adic points of $\mathcal{G}$ determined by $\rho$ for a prime $\mathfrak{p}$ of $\mathfrak{K}$. We prove that the local zeta functions are almost uniform for all $\mathfrak{K}$-split groups whose unipotent radical satisfies a certain lifting property. This property is automatically satisfied if $\mathcal{G}$ is reductive. We provide a further criterion in terms of invariants of $\mathcal{G}$ and $\rho$ which guarantees that the local zeta functions satisfy functional equations for almost all primes of $\mathfrak{K}$. We obtain these results by using a $\mathfrak{p}$-adic Bruhat decomposition of Iwahori and Matsumoto [IM] to express the zeta function as a weighted sum over the Weyl group $W$ associated to $\mathcal{G}$ of generating functions over lattice points of a polyhedral cone. The functional equation reflects an interplay between symmetries of the Weyl group and reciprocities of the combinatorial object. We construct families of groups with representations violating our second structural criterion whose local zeta functions do not satisfy functional equations. Our work generalizes results of Igusa [Igu] and du Sautoy and Lubotzky [dSL] and has implications for zeta functions of finitely generated torsion-free nilpotent groups.Comment: 22 page

Koszul dual 2-functors and extension algebras of simple modules for GL2GL_2

Let p be a prime number. We compute the Yoneda extension algebra of $GL_2$ over an algebraically closed field of characteristic p by developing a theory of Koszul duality for a certain class of 2-functors, one of which controls the category of rational representations of $GL_2$ over such a field.Comment: 39 pages, title changed in second version, to appear in Selecta Math. (N.S.

Non-commutative Noetherian Unique Factorisation Domains.

The commutative theory of Unique Factorisation Domains (UFDs) is well-developed (see, for example, Zariski¬-Samuel[75], Chapter 1, and Cohn[2l], Chapter 11). This thesis is concerned with classes of non-commutative Noetherian rings which are generalisations of the commutative idea of UFD. We may characterise commutative Unique Factorisation Domains amongst commutative domains as those whose height-l prime ideals P are all principal (and completely prime ie R/P is a domain). In Chatters[l3], A.W.Chatters proposed to extend this definition to non-commutative Noetherian domains by the simple expedient of deleting the word commutative from the above. In Section 2.1 we describe the definition and some of the basic theory of Noetherian UFDs, and in Sections 2.2, 2.3, and 2.4 demonstrate that large classes of naturally occuring Noetherian rings are in fact Noetherian UFDs under this definition. Chapter 3 develops some of the more surprising consequnces of the theory by indicating that if a Noetherian UFD is not commutative then it has much better properties than if it were. All the work, unless otherwise indicated, of this Chapter is original and the main result of Section 3.1 appears Gilchrist-Smith[30]. In the consideration of Unique Factorisation Domains the set C of elements of a UFD R which are regular modulo all the height-l prime ideals of R plays a crucial role, akin to that of the set of units in a commutative ring. The main motivation of Chapter 4 has been to generalise the commutative principal ideal theorem to non-commutative rings and so to enable us to draw conclusions about the set C. We develop this idea mainly in relation to two classes of prime Noetherian rings namely PI rings and bounded maximal orders. Chapter 5 then returns to the theme of unique factorisation to consider firstly a more general notion to that of UFD, namely that of Unique Factorisation Ring (UFR) first proposed by Chatters-Jordan[17]. In Section 5.2 we prove some structural results for these rings and in particular an analogue of the decomposition R -snT for R a UFD. Finally section 5.3 briefly sketches two other variations on the theme of unique factorisation due primarily to Cohn[ 20], and Beauregard [4], and shows that in general these theories are distinct

Logical Localism in the Context of Combining Logics

[eng] Logical localism is a claim in the philosophy of logic stating that different logics are correct in different domains. There are different ways in which this thesis can be motivated and I will explore the most important ones. However, localism has an obvious and major challenge which is known as ‘the problem of mixed inferences’. The main goal of this dissertation is to solve this challenge and to extend the solution to the related problem of mixed compounds for alethic pluralism. My approach in order to offer a solution is one that has not been considered in the literature as far as I am aware. I will study different methods for combining logics, concentrating on the method of juxtaposition, by Joshua Schechter, and I will try to solve the problem of mixed inferences by making a finer translation of the arguments and using combination mechanisms as the criterion of validity. One of the most intriguing aspects of the dissertation is the synergy that is created between the philosophical debate and the technical methods with the problem of mixed inferences at the center of that synergy. I hope to show that not only the philosophical debate benefits from the methods for combining logics, but also that these methods can be developed in new and interesting ways motivated by the philosophical problem of mixed inferences. The problem suggests that there are relevant interactions between connectives, justified by the philosophical considerations for conceptualising different logic systems, that the methods for combining logics should allow to emerge. The recognition of this fact is what drives the improvements on the method of juxtaposition that I develop. That is, in order to allow for the emergence of desirable interaction principles, I will propose alternative ways of combining logic systems -specifically classical and intuitionistic logics- that go beyond the standard for combinations, which is based on minimality conditions so as to avoid the so-called collapse theorems.[spa] El localismo lógico es una tesis en filosofía de la lógica según la cual diferentes sistemas lógicos son correctos en función del dominio en el que se aplican. Dicha tesis cuenta, prima facie, con cierta plausibilidad y con varios argumentos que la respaldan como mostraré. Sin embargo, el localismo se presta a un evidente y poderoso contraargumento conocido como ‘el problema de las inferencias mixtas’. El objetivo principal de esta disertación es dar respuesta a ese problema y extender la solución al problema afín de los compuestos mixtos que afecta al pluralismo alético. La manera de abordar el problema de las inferencias mixtas consistirá en analizar casos paradigmáticos en la literatura a la luz de los métodos de combinación de lógicas. En concreto, me centraré en el método de la yuxtaposición, desarrollado por Joshua Schechter. Así, ofreceré una solución al problema de las inferencias mixtas que pasará por realizar un análisis más sutil y una formalización más precisa de las mismas, para después aplicar los mecanismos de combinación como criterio de validez. Además, mostraré que el problema de las inferencias mixtas provee de multitud de ejemplos que invitan a desarrollar los métodos de combinación de lógicas de formas novedosas. Una de las aportaciones más relevantes de la disertación consistirá en modificar el método de la yuxtaposición para obtener mecanismos que van más allá del estándar de las extensiones mínimas conservativas. En concreto, propondré diferentes mecanismos para combinar la lógica clásica y la intuicionista, de manera que se permita la aparición de distintos principios puente para los que tenemos buenas razones que los justifican, sin que ello conduzca al colapso de las lógicas que se combinan