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.


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