Class numbers of cyclotomic function fields

Abstract

Let q be a prime power and let Fq be the finite field with q elements. For each polynomial Q(T) in lFq[T], one could use the mo(lule to construct an abelian extension of 1Fq(T), called a cyclotomic extension. cyclotomic extensions play a fundamental role in the study of abelian extensions of 1Fq(T), similar to the role played by cyclotomic numrLber fields for abelian extensions of Q. We are interested in the tower of cyclotomic extensions corresponding to the powers of a fixed irreducible polynomial in 1Fq[T]. Two types of properties are obtained for the 1-parts of the class numbers of the fields in this tower, for a fixed prime number 1. One gives congruence relations between the I-parts of these class numbers. The otsher gives lower bound for the 1-parts of these class numbers. Systematic study of cyclotomic field extensions of rational numbers started in the nineteenth century with Kummer and was essential in his work on Fermat's Last Theorem. Towers of cyclotomic number fields were first investigated by Iwasawa in the mid 1950's. One major application of his theory is to determine the growth of the p-divisibility of the class numbers for the fields in the tower [1w]. The study of the cyclotomic theory of function fields started with [Ca] in 1930. Let p be a prime and let q be a power of p. regarded the rational function field k IFq (T) and the associated polynomial ring A -Fq [T] as analogs of the rational number field Q and its ring of integers Z. He constructed an Amodule, later called the Carlitz , out of the completion of the algebraic closure of IFq((T)), an analog of the field of complex numbers. For each polynomial P in A, one could use the module to construct a field extension k(P) of k. The extensions obtained this way are called cyclotomic extensions and are essential in the study of all abelian extensions of k. Fix an irreducible polynomial P in A, and let n run through the set of positive integers. The cyclotomic extensions of k associated to Pn via the module form a tower of extensions: nk c k(P) C k(p2) C .. C k(Pn) C It would be interesting to study the growth of the p-divisibility of the the class numbers for these fields along this tower, as Iwasawa did for cyclotomic extensions of a number field. This is the problem we would like to investigate in this paper. As in the case of a cyclotomic number field, one can decompose the class number h(k(Pn)) of k(Pn) into two integer factors h+(k(Pn)) and h-(k(PT)), called the real part and the relative part of the class number. Let p be the unique prime Received by the editors May 15, 1997. 1991 Mathematics Subject Classification. Primary 1IR29, 11R58; Secondary 11R23.