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Thesis (Ph. D.)--Oregon State University, 2009. / Printout. Includes bibliographical references (leaves 62-64). Also available on the World Wide Web.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1992. / Includes bibliographical references (p. 144-145).

Here we offer an introduction to the adele ring over the field of rational numbers Q and highlight some of its beautiful algebraic and topological structure. We then apply this rich structure to revisit some ancient results of number theory and place them within this modern context as well as make some new observations. We conclude by indicating how this theory enables us to extend the basic arithmetic of Q to the more subtle, complicated, and interesting setting of an arbitrary number field.

For the case of compact groups G = Π∞ j=l Z(p)j which are direct products of countably many copies of a cyclic group of prime order p, links are established between the theories of uniqueness and spectral synthesis on the one hand, and the theory of algebraic numbers on the other, similar to the well-known results of Salem, Meyer et al on the circle. Let p ≥ 2 be a prime and let k{x⁻¹} denote the p-series field of formal Laurent series z = Σhj=₋∞ ajxj with coefficients in the field k = {0, 1,…, p-1} and the integer h arbitrary. Let L(z) = - ∞ if aj = 0 for all j; otherwise let L(z) be the largest index h for which ah ≠ 0. We examine compact sets of the form [Algebraic equation omitted] where θ ε k{x⁻¹}, L(θ) > 0, and I is a finite subset of k[x]. If θ is a Pisot or Salem element of k{x⁻¹}, then E(θ,I) is always a set of strong synthesis. In the case that θ is a Pisot element, more can be proved, including a version of Bochner's property leading to a sharper statement of synthesis, provided certain assumptions are made on I (e.g., I ⊃ {0,1,x,...,xL(θ)-1}). Let G be the compact subgroup of k{x⁻¹} given by G = {z: L(z) < 0}. Let θ ɛ k{x⁻¹}, L(θ) > 0, and suppose L(θ) > 1 if p = 3 and L(θ) > 2 if p = 2. Let I = {0,1,x,...,x²L(θ)-1}. Then E = θ⁻¹Ε(θ,I) is a perfect subset of G of Haar measure 0, and E is a set of uniqueness for G precisely when θ is a Pisot or Salem element. Some byways are explored along the way. The exact analogue of Rajchman's theorem on the circle, concerning the formal multiplication of series, is obtained; this is new, even for p = 2. Other examples are given of perfect sets of uniqueness, of sets satisfying the Herz criterion for synthesis, and sets of multiplicity, including a class of M-sets of measure 0 defined via Riesz products which are residual in G. In addition, a class of perfect M₀-sets of measure 0 is introduced with the purpose of settling a question left open by W.R. Wade and K. Yoneda, Uniqueness and quasi-measures on the group of integers of a p-series field, Proc. A.M.S. 84 (1982), 202-206. They showed that if S is a character series on G with the property that some subsequence {SpNj} of the pn-th partial sums is everywhere pointwise bounded on G, then S must be the zero series if SpNj → 0 a.e.. We obtain a strong complement to this result by establishing that series S on G exist for which Sn → 0 everywhere outside a perfect set of measure 0, and for which sup |SpN| becomes unbounded arbitrarily slowly. / Science, Faculty of / Mathematics, Department of / Graduate

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Let R(G) denote the category of smooth representations of a p-adic group. Bernstein has constructed an indexing set B(G) such that R(G) decomposes into a direct sum over s ∈ B(G) of full subcategories Rs(G) known as Bernstein subcategories. Bushnell and Kutzko have developed a method to study the representations contained in a given subcategory. One attempts to associate to that subcategory a smooth irreducible representation (τ,W) of a compact open subgroup J < G. If the functor V ↦ HomJ(W,V) is an equivalence of categories from Rs(G) → H(G,τ)mod we call (J,τ) a type. Given a Levi subgroup L < G and a type (JL, τL) for a subcategory of representations on L, Bushnell and Kutzko further show that one can construct a type on G that “lies over” (JL, τL) by constructing an object known as a cover. In particular, a cover implements induction of H(L,τL)-modules in a manner compatible with parabolic induction of L-representations. In this thesis I construct a cover for certain representations of the Siegel Levi subgroup of Sp(2k) over an archimedean local field of characteristic zero. In partic- ular, the representations I consider are twisted by highly ramified characters. This compliments work of Bushnell, Goldberg, and Stevens on covers in the self-dual case. My construction is quite concrete, and I also show that the cover I construct has a useful property known as splitness. In fact, I prove a fairly general theorem characterizing when covers are split.