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Lower bounds for relative class numbers of CM-fields. (English) Zbl 0795.11058

Summary: Let \(K\) be a CM-field that is a quadratic extension of a totally real number field \(k\). Under a technical assumption, we show that the relative class number of \(K\) is large compared with the absolute value of the discriminant of \(K\), provided that the Dedekind zeta function of \(k\) has a real zero \(s\) such that \(0<s<1\). This result will enable us to get sharp upper bounds on conductors of totally imaginary abelian number fields with class number one or with prescribed ideal class groups.

MSC:

11R29 Class numbers, class groups, discriminants
11R42 Zeta functions and \(L\)-functions of number fields
11R20 Other abelian and metabelian extensions
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[1] Kenneth Hardy, Richard H. Hudson, David Richman, and Kenneth S. Williams, Determination of all imaginary cyclic quartic fields with class number 2, Trans. Amer. Math. Soc. 311 (1989), no. 1, 1 – 55. · Zbl 0678.12003
[2] Jeffrey Hoffstein, Some analytic bounds for zeta functions and class numbers, Invent. Math. 55 (1979), no. 1, 37 – 47. · Zbl 0474.12009 · doi:10.1007/BF02139701
[3] K. Horie and M. Horie, On the exponent of the class group of \( CM\)-fields, Lecture Notes in Math., vol. 1434, Springer-Verlag, Berlin and New York, 1990, pp. 143-148. · Zbl 0702.11074
[4] S. Lang, Functional equation of the zeta function, Hecke’s proof, Algebraic Number Theory, Graduate Texts in Math., vol. 110, Springer-Verlag, New York.
[5] S. Louboutin, Détermination des corps quartiques cycliques totalement imaginaires à groupes des classes d’idéaux d’exposant \( \leqslant 2\), C. R. Acad. Sci. Paris. Sér. I Math. 315 (1992), 251-254; Manuscripta Math. 77 (1992), 385-404. · Zbl 0782.11028
[6] A. Mallik, A note on J. B. Friedlander’s paper: ”On the class numbers of certain quadratic extensions” [Acta Arith. 28 (1975/76), no. 4, 391 – 393; MR 52 #10683], Acta Arith. 35 (1979), no. 1, 54 – 55.
[7] J. Myron Masley and Hugh L. Montgomery, Cyclotomic fields with unique factorization, J. Reine Angew. Math. 286/287 (1976), 248 – 256. · Zbl 0335.12013
[8] W. Narkiewicz, Elementary and analytic theory of algebraic numbers, PWN, Warsaw, 1976. · Zbl 0333.10035
[9] J. Barkley Rosser, Real roots of real Dirichlet \?-series, J. Research Nat. Bur. Standards 45 (1950), 505 – 514.
[10] H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math. 23 (1974), 135 – 152. · Zbl 0278.12005 · doi:10.1007/BF01405166
[11] Kôji Uchida, Imaginary abelian number fields of degrees 2^{\?} with class number one, Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata, 1986) Nagoya Univ., Nagoya, 1986, pp. 151 – 170. · Zbl 0612.12011
[12] L. C. Washington, Cyclotomic fields of class number one, Introduction to cyclotomic fields, Graduate Texts in Math., vol. 83, Springer-Verlag, New York, 1982. · Zbl 0484.12001
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