  Hyperdeterminants are generalizations of determinants from matrices to
multi-dimensional hypermatrices. They were discovered in the 19th century by
Arthur Cayley but were largely ignored over a period of 100 years before once
again being recognised as important in algebraic geometry, physics and number
theory. It is shown that a cubic elliptic curve whose Mordell-Weil group
contains a Z2 x Z2 x Z subgroup can be transformed into the degree four
hyperdeterminant on a 2x2x2 hypermatrix comprising its variables and
coefficients. Furthermore, a multilinear problem defined on a 2x2x2x2
hypermatrix of coefficients can be reduced to a quartic elliptic curve whose
J-invariant is expressed in terms of the hypermatrix and related invariants
including the degree 24 hyperdeterminant. These connections between elliptic
curves and hyperdeterminants may have applications in other areas including
physics.
