Proceedings of the Japan Academy, Series A, Mathematical Sciences

An inequality between class numbers and Ono's numbers associated to imaginary quadratic fields

Fumio Sairaiji and Kenichi Shimizu

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Abstract

Ono's number $p_D$ and the class number $h_D$, associated to an imaginary quadratic field with discriminant $-D$, are closely connected. For example, Frobenius-Rabinowitsch Theorem asserts that $p_D = 1$ if and only if $h_D = 1$. In 1986, T. Ono raised a problem whether the inequality $h_D \leq 2^{p_D}$ holds. However, in our previous paper [8], we saw that there are infinitely many $D$ such that the inequality does not hold. In this paper we give a modification to the inequality $h_D \leq 2^{p_D}$. We also discuss lower and upper bounds for Ono's number $p_D$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 78, Number 7 (2002), 105-108.

Dates
First available in Project Euclid: 23 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.pja/1148392629

Digital Object Identifier
doi:10.3792/pjaa.78.105

Mathematical Reviews number (MathSciNet)
MR1930211

Zentralblatt MATH identifier
1052.11070

Subjects
Primary: 11R11: Quadratic extensions
Secondary: 11R29: Class numbers, class groups, discriminants

Keywords
Ono's number class number

Citation

Sairaiji, Fumio; Shimizu, Kenichi. An inequality between class numbers and Ono's numbers associated to imaginary quadratic fields. Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 7, 105--108. doi:10.3792/pjaa.78.105. https://projecteuclid.org/euclid.pja/1148392629


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