Vol. 12, No. 9, 2018

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

Tom Fisher

Vol. 12 (2018), No. 9, 2123–2150
Abstract

We extend the formulae of classical invariant theory for the Jacobian of a genus one curve of degree n 4 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.

Keywords
elliptic curves, invariant theory, higher secant varieties
Mathematical Subject Classification 2010
Primary: 11G05
Secondary: 13D02, 14H52
Milestones
Received: 30 August 2017
Revised: 15 June 2018
Accepted: 15 July 2018
Published: 21 December 2018
Authors
Tom Fisher
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Wilberforce Road, Cambridge, CB3 0WB
United Kingdom