Statistics of genus numbers of cubic fields
Annales de l'Institut Fourier, Online first, 19 p.

We prove that approximately 96.23% of cubic fields, ordered by discriminant, have genus number one, and we compute the exact proportion of cubic fields with a given genus number. We also compute the average genus number. Finally, we show that a positive proportion of totally real cubic fields with genus number one fail to be norm-Euclidean.

Nous prouvons qu’il y a approximativement 96.23% de corps de nombres cubiques, ordonnés par discriminant, dont le nombre de genres est un et nous calculons la proportion exacte des corps de nombres cubiques avec un nombre de genres donné. Nous calculons également le nombre de genres moyen. Finalement, nous montrons qu’il y a une proportion strictement positive de corps de nombres cubiques totalement réels avec un nombre de genres un qui ne sont pas euclidiens pour la norme.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3558
Classification: 11R16,  11R29,  11R45
Keywords: cubic field, genus number, discriminant density, norm-Euclidean
McGown, Kevin J. 1; Tucker, Amanda 2

1 California State University, Chico (USA)
2 University of Rochester (USA)
@unpublished{AIF_0__0_0_A124_0,
     author = {McGown, Kevin J. and Tucker, Amanda},
     title = {Statistics of genus numbers of cubic fields},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2022},
     doi = {10.5802/aif.3558},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - McGown, Kevin J.
AU  - Tucker, Amanda
TI  - Statistics of genus numbers of cubic fields
JO  - Annales de l'Institut Fourier
PY  - 2022
DA  - 2022///
PB  - Association des Annales de l’institut Fourier
N1  - Online first
UR  - https://doi.org/10.5802/aif.3558
DO  - 10.5802/aif.3558
LA  - en
ID  - AIF_0__0_0_A124_0
ER  - 
%0 Unpublished Work
%A McGown, Kevin J.
%A Tucker, Amanda
%T Statistics of genus numbers of cubic fields
%J Annales de l'Institut Fourier
%D 2022
%I Association des Annales de l’institut Fourier
%Z Online first
%U https://doi.org/10.5802/aif.3558
%R 10.5802/aif.3558
%G en
%F AIF_0__0_0_A124_0
McGown, Kevin J.; Tucker, Amanda. Statistics of genus numbers of cubic fields. Annales de l'Institut Fourier, Online first, 19 p.

[1] Belabas, Karim A fast algorithm to compute cubic fields, Math. Comput., Volume 66 (1997) no. 219, pp. 1213-1237 | MR

[2] Belabas, Karim; Bhargava, Manjul; Pomerance, Carl Error estimates for the Davenport-Heilbronn theorems, Duke Math. J., Volume 153 (2010) no. 1, pp. 173-210 | DOI | MR

[3] Bhargava, Manjul Higher composition laws. III. The parametrization of quartic rings, Ann. Math., Volume 159 (2004) no. 3, pp. 1329-1360 | DOI | MR

[4] Bhargava, Manjul The density of discriminants of quartic rings and fields, Ann. Math., Volume 162 (2005) no. 2, pp. 1031-1063 | DOI | MR

[5] Bhargava, Manjul Higher composition laws. IV. The parametrization of quintic rings, Ann. Math., Volume 167 (2008) no. 1, pp. 53-94 | DOI | MR

[6] Bhargava, Manjul The density of discriminants of quintic rings and fields, Ann. Math., Volume 172 (2010) no. 3, pp. 1559-1591 | DOI | MR

[7] Bhargava, Manjul; Shankar, Arul; Tsimerman, Jacob On the Davenport-Heilbronn theorems and second order terms, Invent. Math., Volume 193 (2013) no. 2, pp. 439-499 | DOI | MR

[8] Bhargava, Manjul; Taniguchi, Takashi; Thorne, Frank Improved error estimates for the Davenport-Heilbronn theorems (2021) (https://arxiv.org/abs/2107.12819)

[9] Cohen, Henri A course in computational algebraic number theory, Graduate Texts in Mathematics, 138, Springer, 1993, xii+534 pages | DOI | MR

[10] Cohen, Henri; Diaz y Diaz, Francisco; Olivier, Michel Enumerating quartic dihedral extensions of , Compos. Math., Volume 133 (2002) no. 1, pp. 65-93 | MR

[11] Cohen, Henri; Lenstra, Hendrik W. Jr. Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) (Lecture Notes in Mathematics), Volume 1068, Springer, 1984, pp. 33-62 | DOI | MR

[12] Cohen, Henri; Martinet, Jacques Class groups of number fields: numerical heuristics, Math. Comput., Volume 48 (1987) no. 177, pp. 123-137 | DOI | MR

[13] Cohn, Harvey The density of abelian cubic fields, Proc. Am. Math. Soc., Volume 5 (1954), pp. 476-477 | MR

[14] Davenport, Harold Euclid’s algorithm in cubic fields of negative discriminant, Acta Math., Volume 84 (1950), pp. 159-179 | MR

[15] Davenport, Harold; Heilbronn, Hans A. On the density of discriminants of cubic fields. II, Proc. R. Soc. Lond., Ser. A, Volume 322 (1971) no. 1551, pp. 405-420 | MR

[16] Egami, Shigeki On finiteness of the numbers of Euclidean fields in some classes of number fields, Tokyo J. Math., Volume 7 (1984) no. 1, pp. 183-198 | DOI | MR

[17] Ellenberg, Jordan S.; Pierce, Lillian B.; Wood, Melanie Matchett On -torsion in class groups of number fields, Algebra Number Theory, Volume 11 (2017) no. 8, pp. 1739-1778 | DOI | MR

[18] Ellenberg, Jordan S.; Venkatesh, Akshay The number of extensions of a number field with fixed degree and bounded discriminant, Ann. Math., Volume 163 (2006) no. 2, pp. 723-741 | DOI | MR

[19] Erdös, Paul; Ko, Chao Note on the Euclidean Algorithm, J. Lond. Math. Soc., Volume S1-13 (1938) no. 1, p. 3 | DOI | MR

[20] Frölich, Albrecht The genus field and genus group in finite number fields. I, II, Mathematika, Volume 6 (1959), p. 40-46, 142–146 | MR

[21] Hasse, Helmut Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage, Math. Z., Volume 31 (1930) no. 1, pp. 565-582 | DOI | MR

[22] Hasse, Helmut Zur Geschlechtertheorie in quadratischen Zahlkörpern, J. Math. Soc. Japan, Volume 3 (1951), pp. 45-51 | MR

[23] Heilbronn, Hans A. On Euclid’s algorithm in cubic self-conjugate fields, Proc. Camb. Philos. Soc., Volume 46 (1950), pp. 377-382 | MR

[24] Ishida, Makoto The genus fields of algebraic number fields, Lecture Notes in Mathematics, 555, Springer, 1976, vi+116 pages | MR

[25] Landau, Edmund Elementary number theory, Chelsea Publishing, 1958, 256 pages (Translated by J. E. Goodman) | MR

[26] Lemmermeyer, Franz The Euclidean algorithm in algebraic number fields, Expo. Math., Volume 13 (1995) no. 5, pp. 385-416 | MR

[27] Leopoldt, Heinrich W. Zur Geschlechtertheorie in abelschen Zahlkörpern, Math. Nachr., Volume 9 (1953), pp. 351-362 | MR

[28] McGown, Kevin J. Norm-Euclidean cyclic fields of prime degree, Int. J. Number Theory, Volume 8 (2012) no. 1, pp. 227-254 | DOI | MR

[29] McGown, Kevin J. Norm-Euclidean Galois fields and the generalized Riemann hypothesis, J. Théor. Nombres Bordeaux, Volume 24 (2012) no. 2, pp. 425-445 | MR | Zbl

[30] McGown, Kevin J.; Thorne, Frank; Tucker, Amanda Counting quintic fields with genus number one (2020) (https://arxiv.org/abs/2006.12991)

[31] Roberts, David P. Density of cubic field discriminants, Math. Comput., Volume 70 (2001) no. 236, p. 1699-1705 (electronic) | DOI | MR

[32] Shankar, Arul; Tsimerman, Jacob Counting S 5 -fields with a power saving error term, Forum Math. Sigma, Volume 2 (2014), e13, 8 pages | DOI | MR

[33] Taniguchi, Takashi; Thorne, Frank Secondary terms in counting functions for cubic fields, Duke Math. J., Volume 162 (2013) no. 13, pp. 2451-2508 | DOI | MR

Cited by Sources: