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Infinite families of irregular primes in cyclotomic function fields
Title
Infinite families of irregular primes in cyclotomic function fields
Authors
Lee J.
;
Lee Y.
Ewha Authors
이윤진
SCOPUS Author ID
이윤진
Issue Date
2020
Journal Title
Journal of Number Theory
ISSN
0022-314X
Citation
Journal of Number Theory vol. 207, pp. 1 - 21
Keywords
Function field
;
Irregular prime
;
Regulator
;
Sextic extension
Publisher
Academic Press Inc.
Indexed
SCIE; SCOPUS
Document Type
Article
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). © 2018
DOI
10.1016/j.jnt.2018.09.008
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