  In the recent years a lot of effort has been made to extend the theory of
hyperholomorphic functions from the setting of associative Clifford algebras to
non-associative Cayley-Dickson algebras, starting with the octonions. An
important question is whether there appear really essentially different
features in the treatment with Cayley-Dickson algebras that cannot be handled
in the Clifford analysis setting. Here we give one concrete example.
Cayley-Dickson algebras namely admit the construction of direct analogues of
CM-lattices, in particular lattices that are closed under multiplication.
Canonical examples are lattices with components from the algebraic number
fields $\mathbb{Q}[\sqrt{m_1},\ldots\sqrt{m_k}]$. Note that the multiplication
of two non-integer lattice paravectors does not give anymore a lattice
paravector in the Clifford algebra. In this paper we exploit the tools of
octonionic function theory to set up an algebraic relation between different
octonionic generalized elliptic functions which give rise to octonionic
elliptic curves. We present formulas for the trace of the octonionic
CM-division values.
