Infinite families of irregular primes in cyclotomic function fields

Jungyun Lee, Yoonjin Lee

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2 Scopus citations

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(Λf) and the Jacobian of k(Λf)F¯q, where f is an irreducible polynomial in the rational function field k=Fq(t) and Fq 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(Λf)+ with conductor f are divisible by N. For the first family of irregular primes, N is equal to pp(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(Λf)+ has p-rank at least p(p−1).

Original languageEnglish
Pages (from-to)1-21
Number of pages21
JournalJournal of Number Theory
Volume207
DOIs
StatePublished - Feb 2020

Bibliographical note

Funding Information:
The authors were supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827), the first named author by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2017R1A6A3A11030486) and the second named author by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2017R1A2B2004574).

Publisher Copyright:
© 2018

Keywords

  • Function field
  • Irregular prime
  • Regulator
  • Sextic extension

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