Function Theories in Cayley-Dickson algebras and Number Theory

TL;DR

This paper explores the extension of hyperholomorphic function theory to Cayley-Dickson algebras, revealing unique features such as the construction of multiplication-closed lattices and their relation to octonionic elliptic functions and curves.

Contribution

It demonstrates the existence of algebraic structures like CM-lattices in Cayley-Dickson algebras and develops octonionic elliptic functions, highlighting features not present in Clifford analysis.

Findings

Construction of multiplication-closed lattices in Cayley-Dickson algebras.

Development of octonionic elliptic functions and curves.

Formulas for trace of octonionic CM-division values.

Abstract

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.