×

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

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.

MSC:

11E76 Forms of degree higher than two
11E25 Sums of squares and representations by other particular quadratic forms
PDFBibTeX XMLCite
Full Text: DOI Link

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)
[8] Gross, B. H., Groups over \(Z\), Invent. Math., 124, 263-278 (1996)
[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
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.