Elliptic Curves and Hyperdeterminants in Quantum Gravity

TL;DR

This paper explores the mathematical relationship between elliptic curves and hyperdeterminants, revealing potential applications in physics and advancing understanding in algebraic geometry and number theory.

Contribution

It demonstrates how elliptic curves with specific Mordell-Weil groups relate to hyperdeterminants and reduces multilinear problems to elliptic curves, highlighting new connections.

Findings

Elliptic curves can be transformed into hyperdeterminants.

Multilinear problems on hypermatrices reduce to elliptic curves.

J-invariant of the elliptic curve relates to hyperdeterminants.

Abstract

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.