Functiones et Approximatio Commentarii Mathematici

Existence of an infinite family of pairs of quadratic fields $\mathbb{Q}(\sqrt{m_1D})$ and $\mathbb{Q}(\sqrt{m_2D})$ whose class numbers are both divisible by $3$ or both indivisible by $3$

Akiko Ito

Full-text: Open access

PDF File (243 KB)

Abstract

Let $m_1$, $m_2$, and $m_3$ be distinct square-free integers (including $1$). First, we show that there exist infinitely many square-free integers $d$ with $\gcd(m_1m_2, d) = 1$ such that the class numbers of $\mathbb{Q}(\sqrt{m_1d})$ and $\mathbb{Q}(\sqrt{m_2d})$ are both divisible by $3$. This is a generalization of a result of T.~Komatsu [15]. Secondly, we show that there exist infinitely many positive fundamental discriminants $D$ with $\gcd(m_1m_2m_3, D) = 1$ such that the class numbers of real quadratic fields $\mathbb{Q}(\sqrt{m_1D})$, $\mathbb{Q}(\sqrt{m_2D})$, and $\mathbb{Q}(\sqrt{m_3D})$ are all indivisible by $3$ when $m_1$, $m_2$, and $m_3$ are positive. This is a generalization of a result of D.~Byeon [4]. We add an application of this result to the Iwasawa invariants related to Greenberg's conjecture.

Article information

Source
Funct. Approx. Comment. Math., Volume 49, Number 1 (2013), 111-135.

Dates
First available in Project Euclid: 20 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.facm/1379686438

Digital Object Identifier
doi:10.7169/facm/2013.49.1.8

Mathematical Reviews number (MathSciNet)
MR3127903

Zentralblatt MATH identifier
1273.11158

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

Keywords
quadratic fields class numbers Iwasawa invariants

Citation

Ito, Akiko. Existence of an infinite family of pairs of quadratic fields $\mathbb{Q}(\sqrt{m_1D})$ and $\mathbb{Q}(\sqrt{m_2D})$ whose class numbers are both divisible by $3$ or both indivisible by $3$. Funct. Approx. Comment. Math. 49 (2013), no. 1, 111--135. doi:10.7169/facm/2013.49.1.8. https://projecteuclid.org/euclid.facm/1379686438


Export citation

References

  • N.C. Ankeny and S. Chowla, On the divisibility of the class number of quadratic fields, Pacific J. Math. 5 (1955), 321–324.
    Mathematical Reviews (MathSciNet): MR85301
    Digital Object Identifier: doi:10.2140/pjm.1955.5.321
    Project Euclid: euclid.pjm/1103044454
  • D. Byeon, Indivisibility of class numbers and Iwasawa $\lambda$-invariants of real quadratic fields, Compositio Math. 126 (2001), no. 3, 249–256.
    Mathematical Reviews (MathSciNet): MR1834738
    Digital Object Identifier: doi:10.1023/A:1017561005538
  • D. Byeon, Existence of certain fundamental discriminants and class numbers of real quadratic fields, J. Number Theory 98 (2003), no. 2, 432–437.
    Mathematical Reviews (MathSciNet): MR1955426
    Digital Object Identifier: doi:10.1016/S0022-314X(02)00050-1
  • D. Byeon, Class numbers of quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{tD})$, Proc. Amer. Math. Soc. 132 (2004), no. 11, 3137–3140.
    Mathematical Reviews (MathSciNet): MR2073286
    Digital Object Identifier: doi:10.1090/S0002-9939-04-07536-7
  • H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1551, 405–420.
    Mathematical Reviews (MathSciNet): MR491593
    Digital Object Identifier: doi:10.1098/rspa.1971.0075
  • B. Ferrero and L.C. Washington, The Iwasawa invariant $\mu _p$ vanishes for abelian number fields, Ann. of Math. (2) 109 (1979), no. 2, 377–395.
    Mathematical Reviews (MathSciNet): MR528968
    Digital Object Identifier: doi:10.2307/1971116
  • R. Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976), no. 1, 263–284.
    Mathematical Reviews (MathSciNet): MR401702
    Digital Object Identifier: doi:10.2307/2373625
  • P. Hartung, Proof of the existence of infinitely many imaginary quadratic fields whose class number is not divisible by $3$, J. Number Theory 6 (1974), 276–278.
    Mathematical Reviews (MathSciNet): MR352040
    Digital Object Identifier: doi:10.1016/0022-314X(74)90022-5
  • K. Horie and Y. Onishi, The existence of certain infinite families of imaginary quadratic fields, J. Reine Angew. Math. 390 (1988), 97–113.
  • K. Horie, A note on basic Iwasawa $\lambda$-invariants of imaginary quadratic fields, Invent. Math. 88 (1987), 31–38.
    Mathematical Reviews (MathSciNet): MR877004
    Digital Object Identifier: doi:10.1007/BF01405089
  • K. Horie, Trace formulae and imaginary quadratic fields, Math. Ann. 288 (1990), 605–612.
    Mathematical Reviews (MathSciNet): MR1081266
    Digital Object Identifier: doi:10.1007/BF01444553
  • K. Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg 20 (1956), 257–258.
    Mathematical Reviews (MathSciNet): MR83013
  • W. Kohnen and K. Ono, Indivisibility of class numbers of imaginary quadratic fields and orders of Tate-Shafarevich groups of elliptic curves with complex multiplication, Invent. Math. 135 (1999), 387–398.
    Mathematical Reviews (MathSciNet): MR1666783
    Digital Object Identifier: doi:10.1007/s002220050290
  • T. Komatsu, A family of infinite pairs of quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{-D})$ whose class numbers are both divisible by $3$, Acta Arith. 96 (2001), 213–221.
    Mathematical Reviews (MathSciNet): MR1814277
    Digital Object Identifier: doi:10.4064/aa96-3-2
  • T. Komatsu, An infinite family of pairs of quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{mD})$ whose class numbers are both divisible by $3$, Acta Arith. 104 (2002), 129–136.
    Mathematical Reviews (MathSciNet): MR1914248
    Digital Object Identifier: doi:10.4064/aa104-2-3
  • P. Llorente and E. Nart, Effective determination of the decomposition of the rational primes in a cubic field, Proc. Amer. Math. Soc. 87 (1983), 579–585.
    Mathematical Reviews (MathSciNet): MR687621
    Digital Object Identifier: doi:10.1090/S0002-9939-1983-0687621-6
  • R.A. Mollin, Solutions of Diophantine equations and divisibility of class numbers of complex quadratic fields, Glasgow Math. J. 38 (1996), 195–197.
    Mathematical Reviews (MathSciNet): MR1397175
    Digital Object Identifier: doi:10.1017/S0017089500031438
  • T. Nagell, Über die Klassenzahl imaginär-quadratischer Zahlkörper, Abh. Math. Sem. Univ. Hamburg 1 (1922), 140–150.
    Mathematical Reviews (MathSciNet): MR3069394
  • J. Nakagawa and K. Horie, Elliptic curves with no rational points, Proc. Amer. Math. Soc. 104 (1988), 20–24.
    Mathematical Reviews (MathSciNet): MR958035
    Digital Object Identifier: doi:10.1090/S0002-9939-1988-0958035-0
  • K. Ono, Indivisibility of class numbers of real quadratic fields, Compositio Math. 119 (1999), no. 1, 1–11.
    Mathematical Reviews (MathSciNet): MR1711515
    Digital Object Identifier: doi:10.1023/A:1001533613223
  • A. Scholz, Über die Beziehung der Klassenzahlen quadratischer Körper zueinander, J. Reine Angew. Math. 166 (1932), 201–203.
  • H. Taya, Iwasawa invariants and class numbers of quadratic fields for the prime $3$, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1285–1292.
    Mathematical Reviews (MathSciNet): MR1641133
    Digital Object Identifier: doi:10.1090/S0002-9939-99-05177-1
  • L.C. Washington, Introduction to Cyclotomic fields, second edition, GTM 83, Springer.
    Mathematical Reviews (MathSciNet): MR1421575
  • P.J. Weinberger, Real quadratic fields with class numbers divisible by $n$, J. Number Theory 5 (1973), 237–241.
    Mathematical Reviews (MathSciNet): MR335471
    Digital Object Identifier: doi:10.1016/0022-314X(73)90049-8
  • Y. Yamamoto, On unramified Galois extentions of quadratic number fields, Osaka J. Math. 7 (1970), 57–76.
    Mathematical Reviews (MathSciNet): MR266898
    Project Euclid: euclid.ojm/1200692686

Adam Mickiewicz University, Faculty of Mathematics and Computer Science

Functiones et Approximatio Commentarii Mathematici

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more