Integral embeddings of cubic norm structures. (English) Zbl 0990.11017

Gan, Wee Teck; Gross, Benedict H.

J. Algebra 233, No. 1, 363-397 (2000).

In a previous paper [B. H. Gross and W. T. Gan, Math. Ann. 314, 265-283 (1999; Zbl 0990.11019)], the authors computed the number of embeddings of the ring of integers in a quadratic or cubic field into various nonassociative rings arising from Coxeter’s order in Cayley’s octonion algebra over \({\mathbb Q}\). The results were obtained by reduction to the local case. The proof of a key proposition in the local case has a gap that is filled in the present paper. Denoting by \(\underline{G}\) the automorphism group of the nonassociative structure over \({\mathbb Z}\) and by \(\underline{H}\) the stabilizer of a global embedding, the authors show that \(\underline{G}({\mathbb Z}_{p})\) acts transitively on the set of embeddings over \({\mathbb Z}_{p}\) and that \(\underline{H}({\mathbb Z}_{p})\) is a special maximal parahoric subgroup of \(\underline{H}({\mathbb Q}_{p})\). In the cubic case, where \(\underline{H}\) is a trialitarian form of \(\text{Spin}_{8}\), they need to extend the results of F. Bruhat and J. Tits [Bull. Soc. Math. Fr. 115, 141-195 (1987; Zbl 0636.20027)] to show that \(\underline{H}({\mathbb Z}_{p})\) is a special maximal compact subgroup.

Reviewer: J.-P.Tignol (Louvain-La-Neuve)

MSC:

11E76	Forms of degree higher than two
11E25	Sums of squares and representations by other particular quadratic forms

Keywords:

group scheme over the integers; parahoric subgroup; triality

Citations:

Zbl 0990.11019; Zbl 0636.20027

References:

[1]	Aschbacher, Chevalley Group of type \(G_2\) as the group of a trilinear form, J. Algebra, 109, 193-259 (1987) · Zbl 0618.20030
[2]	Bourbaki, N., Chap. 9, Algebre (1959), Hermann: Hermann Paris · Zbl 0102.25503
[3]	van der Blij, F.; Springer, T. A., The arithmetics of octaves and the group \(G_2\), Indag. Math., 21, 406-418 (1959) · Zbl 0089.25803
[4]	Bruhat, F.; Tits, J., Groupes réductifs sur un corps local, II, Inst. Hautes Études Sci. Publ. Math., 60, 197-376 (1984)
[5]	Bruhat, F.; Tits, J., Schémas en groupes et immeubles des groupes classiques sur un corps local. II. Groupes unitaires, Bull. Soc. Math. France, 115, 141-195 (1987) · Zbl 0636.20027
[6]	Elkies, N. D.; Gross, B. H., The exceptional cone and the Leech lattice, Internat. Math. Res. Notices, 14, 665-698 (1996) · Zbl 0863.11027
[7]	Grothendieck, A.; Dieudonné, J., EGA 4: Étude locale des schémas et des morphismes de schémas (4th partie), Inst. Hautes Études Sci. Publ. Math., 32, 5-361 (1967) · Zbl 0153.22301
[8]	Gross, B. H., Groups over \(Z\), Invent. Math., 124, 263-278 (1996) · Zbl 0846.20049
[9]	Garibaldi, R., Isotropic trialitarian algebraic groups, J. Algebra, 210, 385-418 (1998) · Zbl 0920.20037
[10]	Gross, B. H.; Gan, W. T., Commutative subrings of certain non-associative rings, Math. Ann., 314, 265-283 (1999) · Zbl 0990.11018
[13]	Jacobson, N., Composition algebras and their automorphisms, Rend. Palermo (1958) · Zbl 0083.02702
[14]	Jacobson, N., Structure and Representations of Jordan Algebras. Structure and Representations of Jordan Algebras, Amer. Math. Soc. Colloq. Pub., 39 (1968), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0218.17010
[15]	Jacobson, N., Exceptional Lie Algebras. Exceptional Lie Algebras, Lecture Notes in Pure and Appl. Math., 1 (1971), Dekker: Dekker New York · Zbl 0215.38701
[16]	Jacobson, N., Some groups of transformations defined by Jordan algebras. II. Groups of type \(F_4\), J. Reine Angew. Math., 204, 74-98 (1960) · Zbl 0142.26401
[17]	Jacobson, N., Structure theory of Jordan algebras, Univ. Arkansas Lecture Notes Math. Sci. (1981), Wiley: Wiley New York · Zbl 0492.17009
[18]	Jacobson, N.; McCrimmon, K., Quadratic Jordan algebras of quadratic forms with base points, J. Indian Math. Soc. (N.S.), 35, 1-45 (1971) · Zbl 0253.17014
[19]	Knus, M.-A.; Merkejev, A.; Rost, M.; Tignol, J.-P., The Book of Involutions. The Book of Involutions, Amer. Math. Sci. Colloq. Pub., 44 (1998), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0955.16001
[20]	Serre, J. P., Résumé des cours 1982-83, Collected Works (1985), Springer-Verlag: Springer-Verlag New York/Berlin
[21]	Springer, T. A., Jordan algebras and algebraic groups, Ergeb. Math. (1973), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0259.17003
[22]	Soda, D., Some groups of type \(D_4\) defined by Jordan algebras, J. Reine Angew Math., 223, 150-163 (1966) · Zbl 0142.26303