Embeddings into the integral octonions. Special issue of the
Pacific Journal of Mathematics.
(English) Zbl 0981.11036

Aschbacher, Michael (ed.) et al., Olga Taussky-Todd: In memoriam. Cambridge, MA: International Press. Pac. J. Math., Spec. Issue, 147-158 (1998).
Let \(R\) be the (essentially unique) maximal order in the \(\mathbb{Q}\)-algebra of octonions. Let \(K\) be an imaginary quadratic field with ring of integers \(A\) and discriminant \(D\) and let \(L(\varepsilon,s)\) be the \(L\)-function of the quadratic Dirichlet character \(\varepsilon: (\mathbb{Z}/ D\mathbb{Z})^*\to \{\pm 1\}\) associated to \(K\). The authors prove that the number of embeddings of \(A\) into \(R\) is \(-252 L(\varepsilon,-2)\). They deliver two proofs, one using theta series and Eisenstein series of half-integral weight, the other using Tamagawa measures. The first generalizes to non-maximal orders \(A\), the second generalizes to the quaternionic case: Let \(B\) be a maximal order in a definite quaternion algebra over \(\mathbb{Q}\) and \(S\) the set of primes which ramify in the quaternion algebra. Then the number of embeddings of \(B\) into \(R\) is \[ 504 \prod_{p\in S} (p^2-1). \] The question of generalization to non-maximal orders \(B\) remains open.
For the entire collection see [Zbl 0889.00012].


11R52 Quaternion and other division algebras: arithmetic, zeta functions
17A35 Nonassociative division algebras