Class numbers of cyclotomic function fields
Li Xin Guo, Linghsueh Shu · Transactions of the American Mathematical Society · 1999
Let q q be a prime power and let F q {\mathbb F}_q be the finite field with q q elements. For each polynomial Q ( T ) Q(T) in F q [ T ] {\mathbb F}_q [T] , one could use the Carlitz module to construct an abelian extension of F q ( T ) {\mathbb F}_q (T) , called a Carlitz cyclotomic extension. Carlitz cyclotomic extensions play a fundamental role in the study of abelian extensions of F q ( T ) {\mathbb F}_q(T) , similar to the role played by cyclotomic number fields for abelian extensions of Q {\mathbb Q} . We are interested in the tower of Carlitz cyclotomic extensions corresponding to the powers of a fixed irreducible polynomial in F q [ T ] {\mathbb F}_q [T] . Two types of properties are obtained for the l l -parts of the class numbers of the fields in this tower, for a fixed prime number l l . One gives congruence relations between the l