On zeros of characteristic p zeta functions

Abstract

The 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.