Journal of Mathematical Sciences, University of Tokyo
(2003), no. 1, pp. 119-138.

Gross' Conjecture for Extensions Ramified Over Three Points of **P**¹,
by Michael Reid

Journal of Mathematical Sciences, University of Tokyo
(2003), no. 1, pp. 119-138.

Abstract

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* = **F**_{q}(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.

Reference

[1]
Benedict H. Gross,
On the values of abelian *L*-functions at *s* = 0 ,
Journal of the Faculty of Science, University of Tokyo,
Section IA. Mathematics
(1988), no. 1, pp. 177-197.
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Updated January 8, 2008.