Complex tori constructed from Cayley-Dickson algebras
Ivona Grzegorczyk, Ricardo Suarez
TL;DR
The paper constructs complex tori $S_{\mathbb{B}_{1,p,q}}$ from tensor products of Cayley–Dickson algebras via lattice quotients, yielding complex dimension $2^{2p+3q}$. It leverages the adjoint action algebra to realize a full-rank endomorphism structure, identifying it with a matrix algebra of size $2^{2p+3q}$ over $\mathbb{C}$. It proves that $\mathrm{End}_{\mathbb{Z}}(S_{\mathbb{B}_{1,p,q}})$ has rank $2^{4p+6q+1}$ and that $S_{\mathbb{B}_{1,p,q}}$ is isogenous to $E^{2^{2p+3q}}$, where $E$ is a CM elliptic curve with $j=1728$. An analytic representation connects these endomorphisms to $\mathrm{End}_{\mathbb{C}}(\mathbb{B}_{1,p,q})$, tying the lattice, adjoint algebra, and CM structure into a concrete algebraic framework.
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 $1728$.