Algebra & Number Theory

A formula for the Jacobian of a genus one curve of arbitrary degree

Tom Fisher

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We extend the formulae of classical invariant theory for the Jacobian of a genus one curve of degree n4 to curves of arbitrary degree. To do this, we associate to each genus one normal curve of degree n, an n×n alternating matrix of quadratic forms in n variables, that represents the invariant differential. We then exhibit the invariants we need as homogeneous polynomials of degrees 4 and 6 in the coefficients of the entries of this matrix.

Article information

Algebra Number Theory, Volume 12, Number 9 (2018), 2123-2150.

Received: 30 August 2017
Revised: 15 June 2018
Accepted: 15 July 2018
First available in Project Euclid: 5 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 13D02: Syzygies, resolutions, complexes 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx]

elliptic curves invariant theory higher secant varieties


Fisher, Tom. A formula for the Jacobian of a genus one curve of arbitrary degree. Algebra Number Theory 12 (2018), no. 9, 2123--2150. doi:10.2140/ant.2018.12.2123.

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