Euler Systems in Global Function Fields

In this paper, the construction of Euler systems of cyclotomic units in a general global function fields is explained. As an application, an analogue of Gras' conjecture in a global function field is proved. 1. I n t r o d u c t i o n and n o t a t i o n s Kolyvagin [7] introduced his remarkable Euler systems to prove important new results on ideal class groups of nmnber felds. This method was further developed by Rubin in [10 12]. Using similar arguments, Fcng and Xu [2] proved a result of the same type about abelian extensions of rational function fields, hi this paper, we extend [2] to a general global function field. Notation is standard if not explained. Specifically, k is a global function field with a finite constant field Fq of q elements and oc is a fixed infinite prime in k of degree do~. A denotes the ring of the flmctions in k which are holomorphie away from oc. Moo is the set of integral ideals of A and Poo is the set of prime ideals of A. ¢ stands for the unit ideal of A. Let k~ be the completion of k at Received April 25, 1995 and in revised form May 4, 2000