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 admit the construction of direct analogues of so-called CM-lattices, in particular, lattices that are closed under multiplication.Canonical examples are lattices with components from the algebraic number fields mathbb{Q}{[sqrt{m1}, ldots sqrt{mk}]}. 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 explicit formulas for the trace of the octonionic CM-division values.
Highlights
There are a number of different possibilities to generalize complex function theory to higher dimensions.One classical and well-established option is to consider functions in several complex variables in Cn where the classical holomorphicity concept is applied separately to each complex variable, see, for example, [8, 15]
From the viewpoint of algebraic geometry, the theory of several complex variables provides the natural setting to study Abelian varieties and curves. Another possibility is offered by Clifford analysis which considers null-solutions to a generalized Cauchy-Riemann operator that are defined in a subset of vectors or
In the recent years one observes a lot of progress in extending the constructions from the setting of associative Clifford algebras to non-associative CayleyDickson algebras, in particular, to the framework of octonions, see, for example, [7, 9, 11, 17, 18, 19, 25]
Summary
There are a number of different possibilities to generalize complex function theory to higher dimensions. The treatment with Cayley-Dickson algebras provides us with a new feature, if we consider these functions in association with a period lattice that has a non-trivial one-sided Cayley-Dickson multiplication, since we do not have such an algebraic structure in the Clifford analysis setting where we are restricted to define the functions on the space of paravectors. We present an explicit algebraic formula to calculate the trace of the octonionic division values of the generalized octonionic regular Weierstraß ℘-function They turn out to be elements of the number field generated by the algebraic elements of the lattice components and the components of the Legendre-constants which still require an algebraic investigation in the future. Explicit formulas for the division values of generalized multiperiodic functions that we developed in this paper might have some interest in this sense, but there are many fundamental open problems that need to be solved in order to be able to give some satisfactory answers to this question
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