On regularity of small primes in function fields

Abstract

Let p be a finite prime of the rational function field K= F q ( T) and K( p ): K the pth cyclotomic extension. We study the p-components of various class groups associated with K( p ), using criteria of Kummer-Herbrand-Ribet type and explicit formulas for Bernoulli-Goss and Bernoulli-Carlitz numbers. A conjecture is formulated that hypothetically gives necessary and sufficient conditions for the non-vanishing of the p-class group of the ring of integers in the maximal “totally real” subfield K + ( p ) of K( p ).