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The 2-Class Tower of \(\mathbb {Q}(\sqrt{-5460})\)

  • Nigel BostonEmail author
  • Jiuya Wang
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 251)

Abstract

The seminal papers in the field of root-discriminant bounds are those of Odlyzko and Martinet. Both papers include the question of whether the field \(\mathbb {Q}(\sqrt{-5460})\) has finite or infinite 2-class tower. This is a critical case that will either substantially lower the best known upper bound for lim inf of root discriminants (if infinite) or else give a counter-example to what is often termed Martinet’s conjecture or question (if finite). Using extensive computation and introducing some new techniques, we give strong evidence that the tower is in fact finite, establishing other properties of its Galois group en route.

2010 Mathematics Subject Classification

11R11 11R37 20D15 

Notes

Acknowledgements

The authors thank Charles Leedham-Green for his comments on the paper. The first author was supported by Simons Foundation Award MSN-179747. The second author was supported by National Science Foundation grant DMS-1301690.

References

  1. 1.
    E.Benjamin, On imaginary quadratic number fields with \(2\)-class group of rank \(4\) and infinite \(2\)-class field tower, Pacific J. Math. 201 (2001), 257–266.MathSciNetCrossRefGoogle Scholar
  2. 2.
    E.Benjamin, On a question of Martinet concerning the \(2\)-class field tower of imaginary quadratic number fields, Ann. Sci. Math. Quebec 26 (1) (2002), 1–13.Google Scholar
  3. 3.
    E.Benjamin, On the \(2\)-class field tower conjecture for imaginary quadratic number fields with \(2\)-class group of rank \(4\), J. Number Theory 154 (2015), 118–143.Google Scholar
  4. 4.
    W.Bosma, J.J.Cannon, and C.Playoust, The Magma algebra system. I. The user language, J.Symbolic Comput. 24 (1997), 235–265.MathSciNetCrossRefGoogle Scholar
  5. 5.
    N.Boston and H.Nover, Computing pro-\(p\) Galois groups, Lecture Notes in Computer Science 4076 (2006), ANTS VII, 1–10.Google Scholar
  6. 6.
    M.R.Bush, Computation of the Galois groups associated to the 2-class towers of some quadratic fields, J. Number Theory 100 (2003), 313–325.Google Scholar
  7. 7.
    J.-M.Fontaine and B.C.Mazur, Geometric Galois representations, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), Series in Number Theory, 1, Int. Press, Cambridge, MA, 41–78.Google Scholar
  8. 8.
    A.Fröhlich, Central extensions, Galois groups, and ideal class groups of number fields, Contemporary Mathematics 24, AMS (1983).Google Scholar
  9. 9.
    E.S.Golod and I.R.Shafarevich, On the class field tower (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 261–272; English translation in AMS Transl. 48 (1965), 91–102.Google Scholar
  10. 10.
    F.Hajir and C.Maire, Tamely ramified towers and discriminant bounds for number fields. II., J. Symbolic Comput. 33 (2002), 415–423.Google Scholar
  11. 11.
    G.Havas, M.F.Newman, and E.A.O’Brien, On the efficiency of some finite groups, Comm. Algebra 32 (2004), 649–656.MathSciNetCrossRefGoogle Scholar
  12. 12.
    H.Koch, Galois theory of \(p\)-extensions, Springer Monographs in Mathematics. Springer-Verlag, Berlin (2002).CrossRefGoogle Scholar
  13. 13.
    C.R.Leedham-Green, The structure of finite \(p\)-groups, J. London Math. Soc. 50 (1) (1994), 49–67.Google Scholar
  14. 14.
    A.Lubotzky and A.Mann, Powerful \(p\)-groups. II. \(p\)-adic analytic groups, J. Algebra 105 (2) (1987), 506–515.Google Scholar
  15. 15.
    J.Martinet, Tours de corps de classes et estimations de discriminants (French), Invent Math 44 (1978), 65–73.MathSciNetCrossRefGoogle Scholar
  16. 16.
    A.Mouhib, Infinite Hilbert \(2\)-class field tower of quadratic number fields, Acta Arith. 145 (3) (2010), 267–272.Google Scholar
  17. 17.
    H.Nover, Computation of Galois groups associated to the \(2\)-class towers of some imaginary quadratic fields with \(2\)-class group \(C_2 \times C_2 \times C_2\), J. Number Theory 129 (1) (2009), 231–245.Google Scholar
  18. 18.
    E.A.O’Brien, The \(p\)-group generation algorithm, J. Symbolic Comput. 9 (1990), 677–698.Google Scholar
  19. 19.
    A.Odlyzko, Lower bounds for discriminants of number fields, Acta Arith. 29 (1976), 275–297.Google Scholar
  20. 20.
    D.J.S.Robinson, A course in the theory of groups, Springer-Verlag, Berlin (1982).CrossRefGoogle Scholar
  21. 21.
    I.R.Shafarevich, Extensions with prescribed ramification points (Russian), IHES Publ. 18 (1963), 71–95; English translation in I.R.Shafarevich, Collected Mathematical Papers, Springer, Berlin (1989).Google Scholar
  22. 22.
    A.Shalev, The structure of finite p-groups: effective proof of the coclass conjectures, Invent. Math. 115 (2) (1994), 315–345.Google Scholar
  23. 23.
    Y.Sueyoshi, Infinite \(2\)-class field towers of some imaginary quadratic number fields, Acta Arith. 113 (3) (2004), 251–257.Google Scholar
  24. 24.
    Y.Sueyoshi, On \(2\)-class field towers of imaginary quadratic number fields, Far East J. Math. Sci. (FJMS) 34 (3) (2009), 329–339.Google Scholar
  25. 25.
    Y.Sueyoshi, On the infinitude of \(2\)-class field towers of some imaginary quadratic number fields, Far East J. Math. Sci. (FJMS) 42 (2) (2010), 175–187.Google Scholar
  26. 26.
    V.Y.Wang, On Hilbert \(2\)-class fields and \(2\)-towers of imaginary quadratic number fields, arXiv:1508.06552.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA

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