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 Title: The zeta function of an order in a general algebra Author(s): Seyfried, Michael David Doctoral Committee Chair(s): Janusz, Gerald Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics Abstract: Zeta functions have been of major importance in algebraic number theory for many years. They are useful (along with L-functions) in obtaining results concerning the asymptotic distribution of ideals in a given class. In 1980 Bushnell and Reiner were able to extend these classical results to the noncommutative case by considering the zeta function of an order in a finite dimensional semisimple Q(or Q$\sb{\rm p}$)-algebra. What happens when the condition of semisimplicity is lifted is addressed in this thesis.Two problems are discussed here: (1) How can the partial zeta function be expressed in terms of the zeta function described by Bushnell and Reiner. (2) What can be done in the case of the total zeta function. How can one deal with the infinite number of isomorphism classes that must exist here.Analytic methods are used to answer the first question and an explicit determination of the abscissa of convergence is obtained. To answer the second question, an inductive method is developed to show how full lattices in an order can be expressed as a sum of full lattices in somewhat simpler pieces (roughly corresponding to a filtration of the Jacobson radical). This information is then used to express the total zeta function as a product of the zeta function of an order in a semisimple algebra with other factors. Issue Date: 1990 Type: Text Language: English URI: http://hdl.handle.net/2142/20978 Rights Information: Copyright 1990 Seyfried, Michael David Date Available in IDEALS: 2011-05-07 Identifier in Online Catalog: AAI9114406 OCLC Identifier: (UMI)AAI9114406
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