On quadratic extensions of number fields and Iwasawa invariants for basic Z3-extensions

Horie, Kuniaki; Kimura, Iwao

doi:10.2969/jmsj/05120387

J. Math. Soc. Japan

1999

DOI: 10.2969/jmsj/05120387

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On quadratic extensions of number fields and Iwasawa invariants for basic Z3-extensions

Kuniaki Horie¹,

Iwao Kimura²

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Cited by 5 publications

(2 citation statements)

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“…Taya [29] improved their result and showed that the value of (1) is ≥ 17/24. Nakagawa-Horie's result is also extended to a more general situation of quadratic extensions over a fixed number field case by Horie and the author [16], and to the fixed function field case by the author [19], using the theory of zeta functions associated to a certain prehomogeneous vector space (the space of binary cubic forms) developed by Datskovsky and Wright [9].…”

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A note on the existence of certain infinite families of imaginary quadratic fields

Kimura

2003

Acta Arith.

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A note on the existence of certain infinite families of imaginary quadratic fields by Iwao Kimura (Toyama)1. Introduction. In this paper, we give a lower bound of the number of certain families of imaginary quadratic fields whose absolute value of discriminants are less than a given number (the main result is the corollary below). We briefly review the investigations concerned with this kind of problems.Let Z be the ring of rational integers, and Q the field of rational numbers. For any rational prime l, we denote by Z l the ring of l-adic integers. If k is an algebraic number field of finite degree over Q, we denote by h(k) the class number of k and by D(k) the discriminant of k. We let S denote the cardinality of any set S. Let (·/·) be the Legendre-Kronecker symbol.Denote by Q − the set of all imaginary quadratic fields and by Q + the set of all real quadratic fields. For any real number X > 0, let Q − (X) = {k ∈ Q − ; −X < D(k)} and Q + (X) = {k ∈ Q + ; D(k) < X}. Note that Q − (X) and Q + (X) are finite sets.Hartung [13] proved that {k ∈ Q − ; 3 h(k)} is an infinite set, and remarked that his method (an application of Kronecker's class number relation) can be applied to the same statement in which 3 is replaced by any odd prime l. The case of l = 3 is also implied in Davenport-Heilbronn's investigation [10,11]

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A note on the existence of certain infinite families of imaginary quadratic fields

Kimura

2003

Acta Arith.

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“…Thus from (2) and a theorem of Davenport and Heilbronn [2], as refined by Horie and Nakagawa Finally, we mention that Horie and Kimura [5] recently showed that there always exist infinitely many totally imaginary quadratic extensions K over a totally real number field F such that λ …”

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Indivisibility of special values of Dedekind zeta functions of real quadratic fields

Byeon

2003

Acta Arith.

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On class numbers of quadratic extensions¶over function fields

Kimura¹

1998

manuscripta mathematica

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On quadratic extensions of number fields and Iwasawa invariants for basic Z3-extensions

Cited by 5 publications

References 17 publications

A note on the existence of certain infinite families of imaginary quadratic fields

A note on the existence of certain infinite families of imaginary quadratic fields

Indivisibility of special values of Dedekind zeta functions of real quadratic fields

On class numbers of quadratic extensions¶over function fields

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