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Markus Kirschmer
One-class genera of exceptional groups over number fields
Journal de théorie des nombres de Bordeaux, 30 no. 3 (2018), p. 847-857, doi: 10.5802/jtnb.1052
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Class. Math.: 20G30, 20G41
Keywords: Class numbers, exceptional groups

Résumé - Abstract

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.


[1] Hans Ulrich Besche, Bettina Eick & Eamonn A. O’Brien, The groups of order at most 2000, Electron. Res. Announc. Am. Math. Soc. 7 (2001), p. 1-4
[2] Armand Borel, Some finiteness properties of adele groups over number fields, Publ. Math., Inst. Hautes Étud. Sci. 16 (1963), p. 5-30
[3] Arjeh Cohen, Gabriele Nebe & Wilhelm Plesken, Maximal integral forms of the algebraic group ${G}_2$ defined by finite subgroups, J. Number Theory 72 (1998), p. 282-308
[4] Benedict H. Gross, Groups over $\mathbb{Z}$, Invent. Math. 124 (1996), p. 263-279
[5] William M. Kantor, Some exceptional $2$-adic buildings, J. Algebra 92 (1985), p. 208-223
[6] William M. Kantor, Robert A. Liebler & Jacques Tits, On discrete chamber-transitive automorphism groups of affine buildings, Bull. Am. Math. Soc. 16 (1987), p. 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. 16 (2013), p. 172-186
[9] Amir Mohammadi & Alireza Salehi Golsefidy, Discrete subgroups acting transitively on vertices of a Bruhat-Tits building, Duke Math. J. 161 (2012), p. 483-544
[10] Takashi Ono, On algebraic groups and discontinuous groups, Nagoya Math. J. 27 (1966), p. 279-322
[11] Gopal Prasad, Volumes of ${S}$-arithmetic quotients of semi-simple groups, Publ. Math., Inst. Hautes Étud. Sci. 69 (1989), p. 91-117
[12] Gopal Prasad & Sai-Kee Yeung, Nonexistence of arithmetic fake compact Hermitian symmetric spaces of type other than ${A}_n$ $(n\le 4)$, J. Math. Soc. Japan 64 (2012), p. 683-731
[13] Tonny A. Springer, Linear algebraic groups, Progress in Mathematics 9, Birkhäuser, 1998
[14] Tonny A. Springer & Ferdinand D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer Monographs in Mathematics, Springer, 2000
[15] Jacques Tits, Reductive groups over local fields, Automorphic forms, representations and ${L}$-functions, Proceedings of Symposia in Pure Mathematics 33, American Mathematical Society, 1979, p. 29–69
[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 5011, Springer, 2008, p. 268–281
[17] George L. Watson, Transformations of a quadratic form which do not increase the class-number, Proc. Lond. Math. Soc. 12 (1962), p. 577-587