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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.

In this paper we deal with a finite abelian group G and the abstract Fourier transform F :defined by the composition with a bijection j : G → Ĝ. ( j is a pullback of j.) In particular, we show that ( j • F ) 2 is a permutation if and only if j is a group isomorphism. Then, we study how the spectra of j • F depends on the isomorphism j.

It has been shown by Madden that there are only finitely many quadratic extensions of k(x), k a finite field, in which the ideal class group has exponent two and the infinity place of k(x) ramifies. We give a characterization of such fields that allow us to find a full list of all such field extensions for future reference. In doing so we correct some errors in earlier published literature.

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