## 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(Λ_{f}) and the Jacobian of k(Λ_{f})F¯_{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(Λ_{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(Λ_{f})^{+} has p-rank at least p(p−1).

Original language | English |
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Pages (from-to) | 1-21 |

Number of pages | 21 |

Journal | Journal of Number Theory |

Volume | 207 |

DOIs | |

State | Published - Feb 2020 |

## Keywords

- Function field
- Irregular prime
- Regulator
- Sextic extension