TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 355, Number 5, Pages 1803–1818 S 0002-9947(03)03245-8 Article electronically published on January 14, 2003 CYCLOTOMIC UNITS AND STICKELBERGER IDEALS OF GLOBAL FUNCTION FIELDS JAEHYUN AHN, SUNGHAN BAE, AND HWANYUP JUNG Abstract. In this paper, we define the group of cyclotomic units and Stickel- berger ideals in any subfield of the cyclotomic function field. We also calculate the index of the group of cyclotomic units in the total unit group in some special cases and the index of Stickelberger ideals in the integral group ring. 1. Introduction In the cyclotomic number field Q(ζ n ), where ζ n = exp(2πi/n), Sinnott [S1] showed that the index of cyclotomic units in the total unit group is equal to the class number of its maximal real subfield up to a simple constant factor (called the Kummer-Sinnott unit-index formula) and the index of the Stickelberger ideal associated to Q(ζ n ) is equal to the relative class number up to a simple constant factor (called the Iwasawa-Sinnott index formula of the Stickelberger ideal). In [S2], he also extended these results to arbitrary abelian number fields over Q. The analogue of Kummer-Sinnott’s unit-index formula, with the Carlitz module assigned to the role played in the classical cyclotomic theory by the multiplicative group, was carried out by Galovich and Rosen [GR] in the rational function field case and by Yin [Y1], replacing the Carlitz module by a general sign-normalized rank- one Drinfeld module in the global function field case. Harrop [Hr] extended the Galovich-Rosen result to any subfield of a cyclotomic function field over the rational function field. The analogue of Iwasawa-Sinnott’s index formula of the Stickelberger ideal was carried out by Yin [Y2], which says that the index of the Stickelberger ideal is equal to the relative ideal class number of the field up to a simple constant factor in the global function field case. Recently Yin [Y3] also defined an ideal (he also called it the Stickelberger ideal) in the integral group ring relative to any finite abelian extension of global fields whose rank is equal to the degree of the extension in the function field case. In (wide or narrow) ray class extension of function fields, he calculated the index of the Stickelberger ideal in the integral group ring, which is equal to the divisor class number up to a simple constant factor. In this article we study the cyclotomic units and the Stickelberger ideal of some abelian extensions of a global function field. Let k be a global function field. Let F/k be a finite abelian extension which is a subfield of a cyclotomic function field. In section 2, we recall some notation and results of a cyclotomic function field over Received by the editors July 1, 2001 and, in revised form, October 28, 2002. 2000 Mathematics Subject Classification. Primary 11R58, 11R60. Key words and phrases. cyclotomic units, Stickelberger ideal, global function field. This work was supported by Korea Research Foundation Grant (KRF-2000-015-DP0010). c 2003 American Mathematical Society 1803 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use