On an analogue of a conjecture of Gross (Q1104979)

In this paper it is studied for some congruence function fields the validity of an analogue to a conjecture of Gross in Iwasawa theory. Let \(K_0\) be an algebraic function field of one variable with field of constants a finite field of characteristic \(p\), \(p\) an odd prime. Let \(K_{\infty}\) be a \(\mathbb{Z}_p\)-extension of \(K_0\), which is either one with no new constants (geometric extension) or a purely constant one (constant extension). Clearly \(G=\mathrm{Gal}(K_{\infty}/K_0)\) operates on \(A\), the \(p\)-part of the \(S\)-class group of \(K_{\infty}\), where \(S\) is a finite set of prime divisors of \(K_0\) (in the geometric case, \(S\) is the set of the ramified primes). The analogue of Gross' conjecture considered states that \(A^G\) is finite. The authors give, in the geometric case, necessary and sufficient conditions in order \(A^G\) to be finite. Using this, they may exhibit examples of fields for which the analogue of Gross' conjecture is true and also examples for which it fails. Moreover, for constant extensions, it is proved that the analogue of Gross' conjecture holds in this case.