Let k be a global function field with the field of constants F q , where q= p n is a prime power. Let be a fixed place with degree d . In the case d =1, we constructed the extended group of cyclotomic units in a cyclotomic extention of (k, ) and calculated its index in the total unit group [Y]. In this paper we extend the results to general d . Surprisingly the formula is independent of d . Here we also give an application of the index formula, which extends and strengthens Shu's Kummer type criterion. Now we state the results precisely.
Let A be the Dedekind subring of k consisting of those functions having no pole other than . Let F (&F q d ) be the residue field at . For a ring R, let R* be the group of invertible elements. Throughout the paper we fix a sign function [Def. 4.1, H3] sgn: k* Γ F* . Let K e be the normalizing field of the triple (k, , sgn), and for a proper ideal m of A let K=K m be the ``cyclotomic'' extension of the triple with conductor m. They arise in the theory of Drinfeld modules. Let H e (resp. H=H m ) be the maximal real subfield of K e (resp. K m ) where splits completely. By [Sects. 15 16, H4], H e is the Hilbert class field of (k, ) and H m is the ray class field of (k, ) modulo m. In this paper, we construct the group E of unramified elliptic units of K e (Def. 2.1) and the extended group C =C m of cyclotomic units of K (Def. 2.5), and calculate their indices in the total unit groups respectively.
Let G 1 =Gal(H e Γk) & Pic(A). Let N=N m be the subgroup of G 1 generated by the Artin symbols { p =(p, H e Γk) for all primes p | m. Let e=[G 1 : N m ] be the index. Let s be the number of distinct prime divisors of m. We regard K e as the cyclotomic extension with conductor e and understand that s=0 means m=e. Let C e =E . For a finite extension FΓk, we denote O F the integral closure of A in F, O* F the unit group of F, h(O F ) the ideal class number of O F , h(F ) the divisor class number of F. Let article no. NT972111 302 0022-314XΓ97 25.00