genus theory for function fields

Bae, Sunghan; Koo, Ja Kyung

doi:10.1017/s1446788700037824

Cited by 20 publications

(29 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The narrow ideal class group Cl + (F ) of F is defined to be the quotient group of fractional ideals modulo principal fractional ideals generated by elements of F + . The narrow genus field G + F of F/k is defined to be the maximal extension of F in H + F which is the compositum of F and some abelian extension of k. See [2] for details on the genus theory of function fields. Let…”

Section: Genus Theory For Function Fieldsmentioning

confidence: 99%

“…In §1 we recall several asymptotic formulas in A = F q [T ], which can be found in [11] and [12]. In §2 we recall the genus theory for function fields [2] and extend some results of Wittmann [13] to the narrow case. In §3.1 we give an asymptotic formula for the number N s,rn of ℓ-cyclic extensions F inside some cyclotomic function fields with ℓ-class number ℓ s and with conductor N of degree rn in the case r > 1.…”

mentioning

confidence: 99%

“…Similar results in the case r = 1 are given in §3. 2. In §4 we give the density for ℓ-ranks in ℓ-cyclic function fields.…”

mentioning

confidence: 99%

See 2 more Smart Citations

ℓ-Ranks of Class Groups of Function Fields

Bae¹,

Jung²

2012

Journal of the Korean Mathematical Society

View full text Add to dashboard Cite

Abstract. In this paper we give asymptotic formulas for the number of ℓ-cyclic extensions of the rational function field k = Fq(T ) with prescribed ℓ-class numbers inside some cyclotomic function fields, and density results for ℓ-cyclic extensions of k with certain properties on the ideal class groups. IntroductionLet Q be the field of rational numbers and ℓ be a prime number. In the 1980s F. Gerth studied extensively the asymptotic behavior of ℓ-cyclic extensions of Q with certain conditions on the ideal class groups and ramified primes. Let us recall Gerth's results more precisely. Write N s,x for the number of ℓ-cyclic extensions of Q with conductor ≤ x and ℓ-class number ℓ s . In [5], it is shown that to obtain an asymptotic formula for N s,x , it suffices to count the number M s+1,x of ℓ-cyclic extensions of Q whose conductor is ≤ x and divisible by exactly s + 1 distinct primes, and whose ℓ-class number is ℓ s . In [6], a matrix M over F ℓ is associated to each ℓ-cyclic extension F of Q with s + 1 ramified primes such that the ℓ-class number of F is ℓ s precisely when rank(M ) = s, and an asymptotic formula for N s,x is given by studying the asymptotic behavior of the number of such matrices. In [8], for ℓ = 2, an effective algorithm for computing the density d t,e (resp. d Let k = F q (T ) be the rational function field over the finite field F q . Let ℓ be a prime number different from the characteristic of k and r be the smallest

show abstract

Section: Genus Theory For Function Fieldsmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

ℓ-Ranks of Class Groups of Function Fields

Bae¹,

Jung²

2012

Journal of the Korean Mathematical Society

View full text Add to dashboard Cite

show abstract

“…In [4], Bae and Koo introduced the genus field of the global function field and studied its general properties. [4]) The genus field G(K) of K is the maximal abelian extension of K in H(K) which is the composite of K and some abelian extension of k. Vol.…”

Section: Definition 21 (See Rosenmentioning

confidence: 99%

“…[4]) The genus field G(K) of K is the maximal abelian extension of K in H(K) which is the composite of K and some abelian extension of k. Vol. 97 (2011) Capitulation problem for global function fields 415…”

Section: Definition 21 (See Rosenmentioning

confidence: 99%

Capitulation problem for global function fields

Li¹,

2011

Arch. Math.

View full text Add to dashboard Cite

Let q be a power of an odd prime number p, k = Fq(t) be the rational function field over the finite field Fq. In this paper, we construct infinitely many real (resp. imaginary) quadratic extensions K over k whose ideal class group capitulates in a proper subfield of the Hilbert class field of K. The proof of the infinity of such fields K relies on an estimation of certain character sum over finite fields. Mathematics Subject Classification (2010). Primary 11R58;Secondary 11A15, 11R37.

show abstract

Class fields, Dirichlet characters, and extended genus fields of global function fields

Rzedowski‐Calderón¹,

Villa‐Salvador²

2023

Mathematische Nachrichten

View full text Add to dashboard Cite

We study the extended genus field of an abelian extension of a rational function field. We follow the definition of Anglès and Jaulent, which uses the class field theory. First, we show that the natural definition of extended genus field of a cyclotomic function field obtained by means of Dirichlet characters is the same as the one given by Anglès and Jaulent. Next, we study the extended genus field of a finite abelian extension of a rational function field along the lines of the study of genus fields of abelian extensions of rational function fields. In the absolute abelian case, we compare this approach with the one given by Anglès and Jaulent.

show abstract

genus theory for function fields

Abstract: We study the genus theory for function fields which is the analogue of the classical genus theory developed by Hasse and Frohlich.

Cited by 20 publications

References 4 publications

ℓ-Ranks of Class Groups of Function Fields

ℓ-Ranks of Class Groups of Function Fields

Capitulation problem for global function fields

Class fields, Dirichlet characters, and extended genus fields of global function fields

Contact Info

Product

Resources

About