Complex tori constructed from Cayley-Dickson algebras

Abstract

In this paper we construct complex tori, denoted by $S_{{\mathbb{B}}_{1,p,q}}$, as quotients of tensor products of Cayley--Dickson algebras, denoted ${\mathbb{B}}_{1,p,q}=\mathbb{C}\otimes {\mathbb{H}}^{\otimes p}\otimes {\mathbb{O}}^{\otimes q}$, with their integral subrings. We then show that these complex tori have endomorphism rings of full rank and are isogenous to the direct sum of $2^{2p+3q}$ copies of an elliptic curve $E$ of $j$-invariant $1728$.