Abstract :
The location and multiplicity of the zeros of zeta functions encode interesting arithmetic information. We study the characteristic p zeta function of Goss. We focus on “trivial” zeros and prove a theorem on zeros at negative integers, 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 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.
