On an analogue of a conjecture of Gross

Abstract

The paper is concerned with a problem in the theory of congruence function fields which is analogous to a conjecture of Gross in Iwasawa Theory. Zp-extensions K∞/K0 of congruence function fields K0 of characteristic p≠2 involving no new constants are considered such that the set S of ramified primes is finite and these primes are fully ramified. Is the set of S-classes invariant under Gal(K∞/K0) finite ? Gross' conjecture asserts that a similar question has an affirmative answer for the class of cyclotomic Zp- extensions of CM-type if S is the set of p-primes and the classes considered are minus S-classes. Using a formula of Witt for the norm residue symbol in cyclic p-extensions of local fields of characteristic p, a necessary and sufficient condition for the validity of the analogue of Gross' conjecture is given for a class of extensions K∞/K0. It is shown by examples that the analogue of Gross' conjecture is not always true.