CM-fields and exponents of their ideal class groups

Horie, Kuniaki; Horie, Mitsuko

doi:10.4064/aa-55-2-157-170

Cited by 18 publications

(8 citation statements)

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“…Let d K , k and d k denote the absolute value of the discriminant of CM number field K, the maximal totally real subfield of K and the discriminant of k, then it is known that h k divides h K , and h − K = h K /h k is called the relative class number of K (see [13]). According to the Brauer-Siegel theorem, if K ranges over CM number fields of a given degree then h − K is asymptotic to log( d K /d k ) and goes to infinity with d K (see [5,Lemma 4]). Can we prove that the exponents of the ideal class groups of the CM number fields of a given degree go to infinity with the absolute values of their discriminants?…”

Section: Introductionmentioning

confidence: 99%

Exponents of the ideal class groups of CM number fields

Louboutin¹,

Okazaki²

2003

Mathematische Zeitschrift

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Section: Introductionmentioning

confidence: 99%

Exponents of the ideal class groups of CM number fields

Louboutin¹,

Okazaki²

2003

Mathematische Zeitschrift

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“…Remarks 17. It is possible to deduce from the usual Brauer-Siegel theorem for class numbers of number fields the following Brauer-Siegel-like result for relative class numbers of normal CM-fields, which improves upon [HH,Lemma 4] (which is given only for CM-fields of a given degree) but is less satisfactory than our previous Theorem 14 (for it is ineffective in the case that N contains no imaginary quadratic subfield):…”

Section: Proof Of Theorem 14mentioning

confidence: 77%

Explicit Lower bounds for residues at 𝑠=1 of Dedekind zeta functions and relative class numbers of CM-fields

Louboutin

2003

Trans. Amer. Math. Soc.

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Abstract. Let S be a given set of positive rational primes. Assume that the value of the Dedekind zeta function ζ K of a number field K is less than or equal to zero at some real point β in the range 1 2 < β < 1. We give explicit lower bounds on the residue at s = 1 of this Dedekind zeta function which depend on β, the absolute value d K of the discriminant of K and the behavior in K of the rational primes p ∈ S. Now, let k be a real abelian number field and let β be any real zero of the zeta function of k. We give an upper bound on the residue at s = 1 of ζ k which depends on β, d k and the behavior in k of the rational primes p ∈ S. By combining these two results, we obtain lower bounds for the relative class numbers of some normal CM-fields K which depend on the behavior in K of the rational primes p ∈ S. We will then show that these new lower bounds for relative class numbers are of paramount importance for solving, for example, the exponent-two class group problem for the non-normal quartic CM-fields. Finally, we will prove Brauer-Siegel-like results about the asymptotic behavior of relative class numbers of CM-fields.

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“…By Proposition 1. (1) (2) The computation consists of two steps : (a) Determine all imaginary cyclic number fields of degree 12 with relative class numbers dividing 16 such that its quartic subfield can be embedded into a field N of type (4 * , 2) with h Mu,Corollary 1], [HH,Lemma 10] or [T, Lemma 1]), we have…”

Section: Resultsmentioning

confidence: 99%

Class numbers of imaginary abelian number fields

Kwon

Chang

2000

Proc. Amer. Math. Soc.

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Abstract. Let N be an imaginary abelian number field. We know that h − N , the relative class number of N , goes to infinity as f N , the conductor of N , approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. It is known that there are only finitely many CMfields N with cyclic ideal class groups of 2-power orders such that the complex conjugation is the square of some automorphism of N . Second, we have proved in this paper that there are exactly 48 such fields.

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CM-fields and exponents of their ideal class groups

Cited by 18 publications

References 0 publications

Exponents of the ideal class groups of CM number fields

Exponents of the ideal class groups of CM number fields

Explicit Lower bounds for residues at 𝑠=1 of Dedekind zeta functions and relative class numbers of CM-fields

Class numbers of imaginary abelian number fields

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