Embeddings into the integral octonions

TLDR: In this paper, the authors show that the number of ring embeddings of A into R lies between 3jDj 5= 2 and 5jDdj 5 = 2.

Abstract: Coxeter discovered a maximal order R inO, which is unique up to the action of Aut(O), with the property that R=pR is an octonion algebra over Z=pZ for all primes p. We review the construction of the order R, and some of its properties, inx1. In x2, we let K be an imaginary quadratic eld, with ring of integers A and discriminant D. We count the number of ring embeddings of A into R, using the L-function L(";s) of the quadratic Dirichlet character " :( Z=DZ)!f 1g associated to K. Theorem 1. The number of embeddings of A into R is 252L("; 2). We give two dierent proofs of this result. The rst uses theta series and Eisenstein series of half-integral weight. The second uses the theory of Tamagawa measures, as developed by Siegel and Weil. From the formula in Theorem 1, it follows that the number of embeddings of A into R lies between 3jDj 5= 2 and 5jDj 5= 2 . Inx3 we let K be a denite quaternion algebra over Q, and let A be a maximal order in K. Let S be the nite set of primes which ramify in K; thus p2 S if and only if KQp is a division algebra over Qp. Using the theory of Tamagawa measures, we will prove the following. Theorem 2. The number of embeddings of A into R is 504 Q p2S (p 2 1). Our interest in octonions dates from a lecture that Serre gave at Harvard on the subject, in the fall of 1990. The embedding problems which we study are generalizations of the results of Hasse and Eichler (cf. [14, p. 92.]) on the embeddings of rings of integers in imaginary quadratic elds into certain orders in rational quaternion algebras. Since Olga loved the arithmetic of quaternion algebras, we felt it was appropriate to dedicate this paper to her memory.