AdvisorThakur, Dinesh S.
MetadataShow full item record
PublisherThe University of Arizona.
RightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
AbstractThe location and multiplicity of the zeros of zeta functions encode interesting arithmetic information. We study characteristic p zeta functions of Carlitz and Goss. We present a simpler proof of the fact that "non-trivial" zeros of a characteristic p zeta function satisfy Goss' analogue of the Riemann Hypothesis for F(q)[T]. We also prove simplicity of these zeros, and give partial results for F(q)[T] where q is not necessarily prime. Then we focus on "trivial" zeros, but for characteristic p zeta functions for general function fields over finite fields. Here, we prove a theorem on zeros at negative integers for characteristic p zeta functions, showing more vanishing than that suggested by naive analogies. We also compute some concrete examples providing the extra vanishing, when the class number is more than one. Finally, we give an application of these results related to the non-vanishing of certain class group components for cyclotomic function fields. In particular, we give examples of function fields, where all the primes of degree more than two are "irregular", in the sense of the Drinfeld-Hayes cyclotomic theory.
Degree ProgramGraduate College