Heuristics for 2-class towers of cyclic cubic fields. (English) Zbl 07815484
Let \(K\) be a cyclic, cubic number field. It is known that the rank \(d\) of the \(2\)-class group \(\text{Cl}_2(K)\) of \(K\) is even and equals the rank of the narrow \(2\)-class group \(\text{Cl}_2^+(K)\). The authors explore the \(2\)-class field tower of \(K\) for the case \(d=2\) by investigating \(G_2(K)\), the Galois group of the maximal unramified \(2\)-extension of \(K\) over \(K\) (and also consider the narrow case). They describe all possible Galois groups up to the fourth layer of the \(2\)-class field tower, give their relative frequencies out of a sample of 500000 fields, and derive some conjectures in the sense of Cohen-Lenstra heuristics. Here it seems that one should focus on the probability of the IPAD of \(G_2(K)\).
Reviewer: Günter Lettl (Graz)
MSC:
| 11R29 | Class numbers, class groups, discriminants |
| 11R37 | Class field theory |
| 11R16 | Cubic and quartic extensions |
References:
| [1] | Armitage, J. V., Fröhlich, A. (1967). Class numbers and unit signatures. Mathematika, 14, 94-98. doi: · Zbl 0149.29501 |
| [2] | Bosma, W., Cannon, J. J., Playoust, C.The Magma algebra system. I. The user language. J. Symbolic Comput. 24, 235-265. doi: · Zbl 0898.68039 |
| [3] | Boston, N. (2007). Galois groups of tamely ramified p-extensions. J. Théorie des Nombres de Bordeaux19(1): 59-70. doi: · Zbl 1123.11038 |
| [4] | Boston, N., Bush, M. R., Hajir, F. (2017). Heuristics for p-class towers of imaginary quadratic fields. Math. Ann. 368(1-2): 633-669. · Zbl 1420.11137 |
| [5] | Boston, N., Bush, M.R., Hajir, F. (2019). Heuristics for p-class towers of real quadratic fields. J. Inst. Math. Jussieu, 20(4): 1-24. |
| [6] | Breen, B., Varma, I., Voight, J. (2019). On unit signatures and narrow class groups of odd abelian number fields: structure and heuristics. Available at: https://arxiv.org/abs/1910.00449, preprint. |
| [7] | Cohen, H., Lenstra, H. W. Jr.(1984). Heuristics on class groups of number fields. In: Number Theory, Noordwijkerhout 1983, LNM 1068, Berlin: Springer, pp. 33-62. · Zbl 0558.12002 |
| [8] | Fontaine, J.-M., Mazur, B. (1995). Geometric Galois representations. In: Coates, J., Yau, S.-T. (eds.), Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), Series in Number Theory, Vol. 1, Cambridge, MA: Int. Press, pp. 41-78. · Zbl 0839.14011 |
| [9] | Garton, D. (2015). Random matrices, the Cohen-Lenstra heuristics, and roots of unity. Algebra Number Theory, 9, 149-171. doi: · Zbl 1326.11068 |
| [10] | Klys, J. (2020). The distribution of p-torsion in degree p cyclic fields. Algebra Number Theory, 14, 815-854. doi: · Zbl 1450.11118 |
| [11] | Koch, H. (2002). “Galois theory of p-extensions.” With a foreword by Shafarevich, I. R.. Translated from the 1970 German original by Franz Lemmermeyer. With a postscript by the author and Lemmermeyer. Springer Monographs in Mathematics. Berlin: Springer-Verlag. |
| [12] | Liu, Y., Wood, M. M., Zureick-Brown, D (2019). A predicted distribution for Galois groups of maximal unramified extensions. Available at: https://arxiv.org/abs/1907.05002. |
| [13] | Malle, G. (2008). Cohen-Lenstra heuristic and roots of unity. J. Number Theory128, 2823-2835. doi: · Zbl 1225.11143 |
| [14] | Malle, G. (2010). On the distribution of class groups of number fields. Exp. Math. 19(4): 465-474. · Zbl 1297.11139 |
| [15] | Malle, G. Tables of cyclic cubic extensions \(K/\mathbb{Q}\) with \(| d_K|\&\#60; 10^{16}\). Available at: https://service.mathematik.uni-kl.de/∼numberfieldtables/KT_3/download.html. |
| [16] | Malle, G., Adam, M. (2015). A class group heuristic based on the distribution of 1-eigenspaces in matrix groups. J. Number Theory149, 225-235. doi: · Zbl 1371.11145 |
| [17] | Mayer, D. C., Ayadi, M., Derhem, A. (2018), p-class field towers of cyclic cubic fields. https://www.researchgate.net/publication/323614009_p-Class_Field_Towers_of_Cyclic_Cubic_Fields. |
| [18] | O’Brien, E. A. (1990). The p-group generation algorithm. J. Symbolic Comput. 9, 677-698. · Zbl 0736.20001 |
| [19] | O’Brien, E. A. (1994). Isomorphism testing for p-groups. J. Symbolic Comput. 17, 133-147. · Zbl 0824.20020 |
| [20] | [The PARI Group, PARI/GP version 2.9.2, Bordeaux, 2017. Available at: http://pari.math.u-bordeaux.fr/. |
| [21] | Rubinstein-Salzedo, S. (2014). Invariants for A4 fields and the Cohen-Lenstra heuristics. Int. J. Number Theory10(5): 1259-1276. doi: · Zbl 1300.11106 |
| [22] | Serre, J.-P. (2007). Topics in Galois Theory, Wellesley, MA: Taylor & Francis. · Zbl 1128.12001 |
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