Gross' Conjecture for Extensions Ramified Over Three Points of P¹

Gross' Conjecture for Extensions Ramified Over Three Points of P¹, by Michael Reid
Journal of Mathematical Sciences, University of Tokyo 10 (2003), no. 1, pp. 119-138.
[JMS] [Math Reviews] [Zentralblatt]
.dvi (69k) .ps (751k) .ps.gz (378k) .pdf (199k)
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.
[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 35 (1988), no. 1, pp. 177-197. [Math Reviews] [Zentralblatt]
Research page Home page E-mail
Updated January 8, 2008.