Quantitative inverse Galois problem for semicommutative finite group schemes
Ratko Darda1 ; Takehiko Yasuda2
1 Department of Mathematics and Computer Science University of Basel Switzerland
2 Department of Mathematics Graduate School of Science Osaka University Japan
Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 1, pp. 105-123.

A semicommutative finite group scheme is a finite group scheme which can be obtained from commutative finite group schemes by iterated performing semidirect products with commutative kernels and taking quotients by normal subgroups. In this article, for an étale tame semicommutative finite group scheme G, we give a lower bound on the number of connected G-torsors of bounded height (such as discriminant).

Un schéma en groupes fini est dit semi-commutatif s’il peut être obtenu à partir de schémas en groupes finis commutatifs en effectuant de manière itérative des produits semi-directs avec des noyaux commutatifs et en prenant des quotients par des sous-groupes normaux. Dans cet article, pour un schéma en groupes fini semi-commutatif étale et modéré G, nous donnons une borne inférieure pour le nombre de G-torseurs connexes de hauteur bornée (tel que le discriminant).

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1314
Classification : 11R45, 11R32, 11R34, 11G50
Keywords: Inverse Galois problem, Malle conjecture, G-torsor, Semiabelian groups

Ratko Darda 1 ; Takehiko Yasuda 2

1 Department of Mathematics and Computer Science University of Basel Switzerland
2 Department of Mathematics Graduate School of Science Osaka University Japan
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{JTNB_2025__37_1_105_0,
     author = {Ratko Darda and Takehiko Yasuda},
     title = {Quantitative inverse {Galois} problem for semicommutative finite group schemes},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {105--123},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {37},
     number = {1},
     year = {2025},
     doi = {10.5802/jtnb.1314},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1314/}
}
TY  - JOUR
AU  - Ratko Darda
AU  - Takehiko Yasuda
TI  - Quantitative inverse Galois problem for semicommutative finite group schemes
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2025
SP  - 105
EP  - 123
VL  - 37
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1314/
DO  - 10.5802/jtnb.1314
LA  - en
ID  - JTNB_2025__37_1_105_0
ER  - 
%0 Journal Article
%A Ratko Darda
%A Takehiko Yasuda
%T Quantitative inverse Galois problem for semicommutative finite group schemes
%J Journal de théorie des nombres de Bordeaux
%D 2025
%P 105-123
%V 37
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1314/
%R 10.5802/jtnb.1314
%G en
%F JTNB_2025__37_1_105_0
Ratko Darda; Takehiko Yasuda. Quantitative inverse Galois problem for semicommutative finite group schemes. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 1, pp. 105-123. doi : 10.5802/jtnb.1314. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1314/

[1] Brandon Alberts Statistics of the first Galois cohomology group: a refinement of Malle’s conjecture, Algebra Number Theory, Volume 15 (2021) no. 10, pp. 2513-2569 | DOI | MR | Zbl

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

[3] Manjul Bhargava The density of discriminants of quintic rings and fields, Ann. Math. (2), Volume 172 (2010) no. 3, pp. 1559-1591 | DOI | MR | Zbl

[4] Nicolas Bourbaki Éléments de mathématique, Masson, 1981, vii+422 pages (Algèbre. Chapitres 4 à 7) | MR | Zbl

[5] Henri Cohen; Anna Morra Counting cubic extensions with given quadratic resolvent, J. Algebra, Volume 325 (2011), pp. 461-478 | DOI | MR | Zbl

[6] Henri Cohen; Frank Thorne Dirichlet series associated to quartic fields with given cubic resolvent, Res. Number Theory, Volume 2 (2016), 29, 40 pages | DOI | MR | Zbl

[7] Henri Cohen; Frank Thorne On Dℓ-extensions of odd prime degree ℓ, Proc. Lond. Math. Soc. (3), Volume 121 (2020) no. 5, pp. 1171-1206 | DOI | MR | Zbl

[8] Jean-Louis Colliot-Thélène; Alexei N. Skorobogatov The Brauer–Grothendieck group, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 71, Springer, 2021, xv+453 pages | DOI | MR | Zbl

[9] Ratko Darda Rational points of bounded height on weighted projective stacks, Ph. D. Thesis, Université Paris Cité (2021) (2021UNIP7094, https://tel.archives-ouvertes.fr/tel-03682761)

[10] Ratko Darda; Takehiko Yasuda Torsors for finite group schemes of bounded height, J. Lond. Math. Soc. (2), Volume 108 (2023) no. 3, pp. 1275-1331 | DOI | MR | Zbl

[11] Ratko Darda; Takehiko Yasuda The Batyrev–Manin conjecture for DM stacks, J. Eur. Math. Soc. (2024) (Online First) | DOI

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

[13] Jordan S. Ellenberg; Matthew Satriano; David Zureick-Brown Heights on stacks and a generalized Batyrev–Manin–Malle conjecture, Forum Math. Sigma, Volume 11 (2023), e14, 54 pages | DOI | MR | Zbl

[14] David Harari Quelques propriétés d’approximation reliées à la cohomologie galoisienne d’un groupe algébrique fini, Bull. Soc. Math. Fr., Volume 135 (2007) no. 4, pp. 549-564 | DOI | Numdam | MR | Zbl

[15] Yonatan Harpaz; Olivier Wittenberg Zéro-cycles sur les espaces homogènes et problème de Galois inverse, J. Am. Math. Soc., Volume 33 (2020) no. 3, pp. 775-805 | DOI | MR | Zbl

[16] Jürgen Klüners A counterexample to Malle’s conjecture on the asymptotics of discriminants, C. R. Math., Volume 340 (2005) no. 6, pp. 411-414 | DOI | MR | Zbl

[17] Gunter Malle On the distribution of Galois groups. II, Exp. Math., Volume 13 (2004) no. 2, pp. 129-135 | DOI | MR | Zbl

[18] Gunter Malle; B. Heinrich Matzat Inverse Galois theory, Springer Monographs in Mathematics, Springer, 1999, xvi+436 pages | DOI | MR | Zbl

[19] James S. Milne Algebraic groups. The theory of group schemes of finite type over a field, Cambridge Studies in Advanced Mathematics, 170, Cambridge University Press, 2017, xvi+644 pages | DOI | MR | Zbl

[20] Jürgen Neukirch Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, 322, Springer, 1999, xviii+571 pages (translated from the 1992 German original and with a note by Norbert Schappacher, with a foreword by G. Harder) | DOI | MR

[21] Jürgen Neukirch; Alexander Schmidt; Kay Wingberg Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften, 323, Springer, 2008, xvi+825 pages | DOI | MR | Zbl

[22] Frans Oort Appendix 3: Questions in arithmetic algebraic geometry, Open problems in arithmetic algebraic geometry (Advanced Lectures in Mathematics), Volume 46, International Press, 2019, pp. 295-331 | MR | Zbl

[23] Jean-Pierre Serre Cohomologie galoisienne, Lecture Notes in Mathematics, 5, Springer, 1994, x+181 pages | DOI | MR | Zbl

[24] David J. Wright Distribution of discriminants of abelian extensions, Proc. Lond. Math. Soc. (3), Volume 58 (1989) no. 1, pp. 17-50 | DOI | MR | Zbl

Cité par Sources :