One-class genera of exceptional groups over number fields
Journal de Théorie des Nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 847-857.

We show that exceptional algebraic groups over number fields do not admit one-class genera of parahoric groups, except in the case G 2 . For the group G 2 , we enumerate all such one-class genera for the usual seven-dimensional representation.

Nous montrons que les groupes algébriques exceptionnels sur un corps de nombres n’admettent pas de genres de groupes parahoriques à une seule classe, sauf dans le cas de G 2 . Pour le groupe G 2 , nous énumérons tous les genres à une seule classe pour la représentation usuelle en dimension 7.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1052
Classification: 20G30,  20G41
Keywords: Class numbers, exceptional groups
Markus Kirschmer 1

1 Lehrstuhl B für Mathematik RWTH Aachen University Pontdriesch 10–16 52062 Aachen, Germany
@article{JTNB_2018__30_3_847_0,
     author = {Markus Kirschmer},
     title = {One-class genera of exceptional groups over number fields},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {847--857},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {3},
     year = {2018},
     doi = {10.5802/jtnb.1052},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1052/}
}
TY  - JOUR
TI  - One-class genera of exceptional groups over number fields
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2018
DA  - 2018///
SP  - 847
EP  - 857
VL  - 30
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1052/
UR  - https://doi.org/10.5802/jtnb.1052
DO  - 10.5802/jtnb.1052
LA  - en
ID  - JTNB_2018__30_3_847_0
ER  - 
%0 Journal Article
%T One-class genera of exceptional groups over number fields
%J Journal de Théorie des Nombres de Bordeaux
%D 2018
%P 847-857
%V 30
%N 3
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.1052
%R 10.5802/jtnb.1052
%G en
%F JTNB_2018__30_3_847_0
Markus Kirschmer. One-class genera of exceptional groups over number fields. Journal de Théorie des Nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 847-857. doi : 10.5802/jtnb.1052. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1052/

[1] Hans Ulrich Besche; Bettina Eick; Eamonn A. O’Brien The groups of order at most 2000, Electron. Res. Announc. Am. Math. Soc., Volume 7 (2001), pp. 1-4 | Zbl: 0986.20024

[2] Armand Borel Some finiteness properties of adele groups over number fields, Publ. Math., Inst. Hautes Étud. Sci., Volume 16 (1963), pp. 5-30 | Zbl: 0135.08902

[3] Arjeh Cohen; Gabriele Nebe; Wilhelm Plesken Maximal integral forms of the algebraic group G 2 defined by finite subgroups, J. Number Theory, Volume 72 (1998) no. 2, pp. 282-308 | Zbl: 0920.11021

[4] Benedict H. Gross Groups over , Invent. Math., Volume 124 (1996), pp. 263-279 | Zbl: 0846.20049

[5] William M. Kantor Some exceptional 2-adic buildings, J. Algebra, Volume 92 (1985), pp. 208-223 | Zbl: 0562.51006

[6] William M. Kantor; Robert A. Liebler; Jacques Tits On discrete chamber-transitive automorphism groups of affine buildings, Bull. Am. Math. Soc., Volume 16 (1987), pp. 129-133

[7] Markus Kirschmer Definite quadratic and hermitian forms with small class number, 2016 (Habilitation thesis, RWTH Aachen University (Germany))

[8] David Lorch; Markus Kirschmer Single-class genera of positive integral lattices, LMS J. Comput. Math., Volume 16 (2013), pp. 172-186 | Zbl: 1297.11018

[9] Amir Mohammadi; Alireza Salehi Golsefidy Discrete subgroups acting transitively on vertices of a Bruhat-Tits building, Duke Math. J., Volume 161 (2012) no. 3, pp. 483-544 | Zbl: 1254.22015

[10] Takashi Ono On algebraic groups and discontinuous groups, Nagoya Math. J., Volume 27 (1966), pp. 279-322 | Zbl: 0166.29802

[11] Gopal Prasad Volumes of S-arithmetic quotients of semi-simple groups, Publ. Math., Inst. Hautes Étud. Sci., Volume 69 (1989), pp. 91-117 | Zbl: 0695.22005

[12] Gopal Prasad; Sai-Kee Yeung Nonexistence of arithmetic fake compact Hermitian symmetric spaces of type other than A n (n4), J. Math. Soc. Japan, Volume 64 (2012), pp. 683-731 | Zbl: 1266.22015

[13] Tonny A. Springer Linear algebraic groups, Progress in Mathematics, Volume 9, Birkhäuser, 1998 | Zbl: 0927.20024

[14] Tonny A. Springer; Ferdinand D. Veldkamp Octonions, Jordan Algebras and Exceptional Groups, Springer Monographs in Mathematics, Springer, 2000 | Zbl: 1087.17001

[15] Jacques Tits Reductive groups over local fields, Automorphic forms, representations and L-functions (Proceedings of Symposia in Pure Mathematics) Volume 33, American Mathematical Society, 1979, pp. 29-69 | Zbl: 0415.20035

[16] John Voight Enumeration of totally real number fields of bounded root discriminant, Algorithmic number theory (ANTS VIII, Banff, 2008) (Lecture Notes in Computer Science) Volume 5011, Springer, 2008, pp. 268-281 | Zbl: 1205.11125

[17] George L. Watson Transformations of a quadratic form which do not increase the class-number, Proc. Lond. Math. Soc., Volume 12 (1962), pp. 577-587 | Zbl: 0107.26901

Cited by Sources: