Jacobians of genus one curves

Sang Yook An, Seog Young Kim, David C. Marshall, Susan H. Marshall, William G. McCallum, Alexander R. Perlis.
Jacobians of genus one curves.
Journal of Number Theory 90 (2001), 304–315.

Consider a curve of genus one over a field K in one of three explicit forms: a double cover of P¹, a plane cubic, or a space quartic. For each form, a certain syzygy from classical invariant theory gives the curve's jacobian in Weierstrass form and the covering map to its jacobian induced by the K-rational divisor at infinity. We give a unified account of all three cases.

Read the MathSciNet review.

2000 Mathematics Subject Classification (MSC):

14H40 Algebraic geometry | Curves | Jacobians
11G05 Number theory | Arithmetic algebraic geometry | Elliptic curves over global fields
14C20 Algebraic geometry | Cycles and subschemes | Divisors, linear systems, invertible sheaves
14C20 Algebraic geometry | Cycles and subschemes | Riemann-Roch theorems
14H52 Algebraic geometry | Curves | Elliptic curves

Key words and phrases:

jacobian
invariant
syzygy
torsion packet
genus one
elliptic curve
hyperosculation
double cover
plane cubic
space quartic

Downloads:

Final article available online at www.idealibrary.com.
Preprint available for download from website of William G. McCallum.
Corrigendum: In second sentence, change "the smallest positive integer" to "a positive integer" (it need not be minimal).

Related work:

For an algorithmic solution using finite group invariant theory, see my doctoral dissertation.
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English translation (version: 12 December 1998) available:
- weil1954a.dvi
- weil1954a.ps
Alfred Clebsch. Über die Wendungsberührebenen der Raumcurven. Journal für die reine und angewandte Mathematik 63 (1864), 1–8.
Annotated English translation (version: 1 November 1998) available:
- clebsch1864.dvi
- clebsch1864.ps
John E. Cremona, Barry Mazur. Visualizing elements in the Shafarevich-Tate group. Experimental Mathematics 9 (2000), 13–28.
Final article available online at www.expmath.org.
Preprint available for download from website of John E. Cremona.
Catherine O'Neil. Jacobians of genus one curves. Mathematical Research Letters 8 (2001), 125–140.
Preprint available for download from website of Catherine O'Neil.
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Preprint available for download from website of Greg W. Anderson.
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Preprint available for download from website of Tom Fisher.