## Elliptic Curves and Hyperdeterminants in Quantum Gravity

**Authors:** Philip Gibbs

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 Z_{2} x Z_{2} 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.

**Comments:** 7 pages. Published in Prespacetime Journal, Vol. 1, Issue 8, pp. 1218-1224 (2010)

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### Submission history

[v1] 29 Sep 2010

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