## Gross' Conjecture for Extensions Ramified over Three Points of $\Bbb P^1$

J. Math. Sci. Univ. Tokyo
Vol. 10 (2003), No. 1, Page 119--138.

Reid, Michael
Gross' Conjecture for Extensions Ramified over Three Points of $\Bbb P^1$
B. Gross has formulated a conjectural generalization of the class number formula. Suppose $L/K$ is an abelian extension of global fields with Galois group $G$. A generalized Stickelberger element $Î¸ \in \ZZ[G]$ is constructed from special values of $L$-functions at $s = 0$. Gross' conjecture then predicts some $I$-adic information about $Î¸$, where $I \subseteq \ZZ[G]$ is the augmentation ideal. In this paper, we prove (under a mild hypothesis) the conjecture for the maximal abelian extension of the rational function field $\FF_q(X)$ that is unramified outside a set of three degree $1$ places.