Circular units of function fields

Abstract

Abstract.A unit index-class number formula is proved for subfields of cyclotomic function fields in analogy with similar results for subfields of cyclotomic number fields.Let m be a positive integer and let C,m = exp{2ni/m).Let Km -Q{Çm) denote the mux cyclotomic field, and F+ its maximal real subfield. There is a parallel setup for function fields of characteristic p. Let ¥q be the finite field with q elements, let RF = ¥q[T] be the ring of polynomials over ¥q (with F transcendental over Fi), and let ¥q{T) be the field of rational functions over ¥q . To each polynomial M e RF one can associate an extension KM, called the Afth cyclotomic function field, which enjoys properties analogous to those of the cyclotomic number field Km . In particular, Galovich and Rosen [1] proved the analogue of Sinnott's theorem in this setting. The purpose of this paper is to extend this unit index-class number formula to an arbitrary subfield of KM.Let k be any subfield of the Afth cyclotomic function field (Af monic), G the Galois group of k over ¥q{T), k+ the maximal subfield of k in which oo splits, Ok{Ok+) the integral closure of ¥q [T] in k{k+), and 0*k the unit group of Ok . In §3, we define a subgroup C of Ok , which we call the circular units of k. Our main result is that C has finite index in Ok , and that this index may be written in the form [Ok':C] = h(Ok+)'4, where h{Ok+) is the class number of Ok+, and ck is a rational number whose definition does not involve h{Ok+).We now briefly describe the contents of the rest of this paper. In §1, we present the relevant definitions and facts in the function field setting. We also state the analytic class number formula. In §2, we review ordinary distributions on ¥q{T)/Rr, discuss an index notation, and obtain a preliminary result on the structure of a certain module. The circular units are introduced in §3 and