Class Numbers of Cyclotomic Function Fields

Let q be a prime power and let Fq be the nite eld with q elements. For each polynomial Q(T) in FqT ], one could use the Carlitz module to construct an abelian extension of Fq(T), called a Carlitz cyclotomic extension. Carlitz cyclotomic extensions play a fundamental role in the study of abelian extensions of Fq(T), similar to the role played by cyclotomic number elds for abelian extensions of Q. We are interested in the tower of Carlitz cyclotomic extensions corresponding to the powers of a xed irreducible polynomial in FqT ]. Two types of properties are obtained for the l-parts of the class numbers of the elds in this tower, for a xed prime number l. One gives congruence relations between the l-parts of these class numbers. The other gives lower bound for the l-parts of these class numbers. Systematic study of cyclotomic eld 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 elds were rst investigated by Iwasawa in the mid 1950's. One major applications of his theory is to determine the growth of the p-divisibility of the class numbers for the elds in the towerrIw]. The study of the cyclotomic theory of function elds started with Carlitz Ca] in 1930. Let p be a prime and let q be a power of p. Carlitz regarded the rational function eld k = F q (T) and the associated polynomial ring A = F q T] as analogs of the rational number eld Q and its ring of integers Z. He constructed a A-module, later called the "Carlitz module" , out of the completion of the algebraic closure of F q ((T)), an analog of the eld of complex numbers. For each polynomial P in A, one could use the Carlitz module to construct a eld 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 P n via the Carlitz module form a tower of extensions: k k(P) k(P 2) k(P n) (0.1) It would be interesting to study the growth of the p-divisibility of the the class numbers for these elds along …