Space-time codes and non-associative division algebras arising from elliptic curves

On the classification of 3-dimensional non-associative division algebras over p-adic fields

“…The classification of n-dimensional non-associative division algebras over fields k has only been accomplished for certain n and k. In the case n = 2, it is known that all non-associative division algebras over k are isotopic to quadratic field extensions of k (see e.g., [12] for references).…”

“…Let us now go through a series of simplifications (as in [11]). Suppose that f (x, y, z) is a cubic anisotropic form over k. If f were reducible over k, then the algebraic set C f defined by the vanishing of f in P 2 would contain a k-rational line and, thus, a k-rational point, a contradiction.…”

“…(1) results on linear matrices and their determinants (Proposition 3.2) derived by the first author from a general result of Beauville and made constructive in his thesis [11] and published in [12] (these results have also been obtained independently in unpublished work of Catherine O'Neil and Manjul Bhargava);…”

“…Recently, non-associative division algebras over number fields have proved useful in constructing "space-time codes," which are matrix codes of complex numbers designed to improve the reliability of radio communications in systems which have more than one transmit antenna (like cell towers). (See [29] for the fundamentals of space-time codes, and [13] for background and references for the application of non-associative division algebras to such codes. )…”

“…In §2 we recall preliminary results on non-associative division algebras, especially the important contributions of Albert and Menichetti. In §3 we recount what we need of [12] on 3-dimensional non-associative division algebras over perfect fields and their classification via their representations. In §4 we apply the arithmetic of elliptic curves over p-adic fields K to reduce the classification of 3-dimensional non-associative division algebras over K to that of classifying ternary cubic forms over K which have no non-trivial zeros over K. In the final §5, we recall what we need of the classical and modern theory of ternary cubic forms, and complete the classification.…”