University of Illinois Urbana-Champaign

The zeta function of an order in a general algebra

Seyfried, Michael David

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Description

Title
The zeta function of an order in a general algebra
Author(s)
Seyfried, Michael David
Issue Date
1990
Doctoral Committee Chair(s)
Janusz, Gerald
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
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.
Type of Resource
text
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http://hdl.handle.net/2142/20978
Copyright and License Information
Copyright 1990 Seyfried, Michael David

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