Infinite families of irregular primes in cyclotomic function fields

Abstract

We find both a lower bound and an upper bound on the p-rank of the divisor class group of the fth cyclotomic function field k(Lambda(f)) and the Jacobian of k(Lambda(f))(F) over bar (q), where f is an irreducible polynomial in the rational function field k = F-q(t) and F-q is the finite field of order q with characteristic p. Moreover, we find two types of infinite families of irregular primes f for which the divisor class numbers of the maximal real cyclotomic function fields k(Lambda(f))(+) with conductor f are divisible by N. For the first family of irregular primes, N is equal to p(p(p-1)), a power of a prime, and for the second family of irregular primes, N is a composite number (pl)(5) for a prime l different from a prime p. Furthermore, in the former case, the divisor class group of k(Lambda(f))(+) has p-rank at least p(p-1). (C) 2018 Published by Elsevier Inc.