Gross [1] has conjectured an algebraic generalization of the usual
analytic class number formula.
In this paper, we prove that Gross' conjecture holds in the following
situation.
Let K = Fq(X) be the
rational function field over the finite field with q elements,
and L the maximal abelian extension of K that is
unramified outside the set of three degree 1 places,
{ 0, 1, ∞ } .
Then, under mild hypotheses, Gross' conjecture holds
for the extension L/K .
The significance of this result is as follows.
Firstly, it gives examples where the genus of the overfield is arbitrarily
large.
Secondly, these examples include (non-constant) cyclic
subextensions of all prime power degrees.
Lastly, the statement of the conjecture in this case predicts a congruence
in a group ring, modulo the cube of the augmentation ideal, which is a
somewhat subtle congruence.