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dc.contributor.advisorVaaler, Jeffrey D.en
dc.creatorPekker, Alexanderen
dc.date.accessioned2008-08-28T23:08:52Zen
dc.date.available2008-08-28T23:08:52Zen
dc.date.issued2006en
dc.identifierb65486973en
dc.identifier.urihttp://hdl.handle.net/2152/2789en
dc.descriptiontexten
dc.description.abstractThis dissertation addresses two problems in diophantine number theory: (1) an analogue of classical Dirichlet’s Theorem in a projective general linear group over a local field and (2) a sharp bound on the conjugate products of successive minima in the geometry of numbers over the adeles. For the first problem, we show that if k is a local field and P GL(N, k) is endowed with a natural norm Φ, then unless A ∈ P GL(N, k) is conjugate to an isometry, the orbit of A, {A, A2 , . . .} is bounded away from the identity element 1N ; otherwise, we quantify min1≤m≤M Φ(Am) in terms of M. We use classical techniques for the case k = C and group-theoretic techniques for k nonarchimedean; as a side fact, we show that the maximal order of an element in P GL(N, Fq) is (q N − 1)/(q − 1). We also discuss a more general vii group-theoretic approach due to Vaaler that is suitable for any compact abelian group and apply it to P GL(N, k). For the second problem, let k be a number field, let A ∈ GL(N, kA) be an automorphism of the adeles kA, and let µn be the n-th absolute successive minimum of A. We show that for 1 ≤ n ≤ N, the conjugate proudct µn(A)µN−n+1(A∗ ) is bounded above by e (N−1)/2 where A∗ = (AT ) −1 is the dual of A. As a corollary, we deduce the (already known) Absolute Siegel’s Lemma, i.e., the fact that QN n=1 µn(A) ≤ e N(N−1)/4 . We use induction, a symmetric algebra argument due to Roy and Thunder, and new duality arguments to derive our results.
dc.format.mediumelectronicen
dc.language.isoengen
dc.rightsCopyright is held by the author. Presentation of this material on the Libraries' web site by University Libraries, The University of Texas at Austin was made possible under a limited license grant from the author who has retained all copyrights in the works.en
dc.subject.lcshDiophantine approximationen
dc.subject.lcshLinear algebraic groupsen
dc.subject.lcshAbelian groupsen
dc.subject.lcshAlgebraic fieldsen
dc.titleDirichlet's Theorem in projective general linear groups and the Absolute Siegel's Lemmaen
dc.description.departmentMathematicsen
dc.identifier.oclc156912025en
dc.type.genreThesisen
thesis.degree.departmentMathematicsen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorThe University of Texas at Austinen
thesis.degree.levelDoctoralen
thesis.degree.nameDoctor of Philosophyen


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