Jacobians of genus one curves
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 P^{1}, 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 Krational divisor at infinity. We give a unified account of all three cases.
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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  RiemannRoch 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.

Remarques sur un mémoire d'Hermite.
Arch. Math.
V (1954),
197–202.
English translation (version: 12 December 1998) available:
.

Über die Wendungsberührebenen der Raumcurven.
Journal für die reine und angewandte Mathematik
63 (1864),
1–8.
Annotated English translation (version: 1 November 1998) available:
.

Visualizing elements in the ShafarevichTate 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.
,
.

Jacobians of genus one curves.
Mathematical Research Letters
8 (2001),
125–140.
Preprint available for download from website of Catherine O'Neil.
.

Abeliants and their application to an elementary construction of Jacobians.
Advances in Mathematics
(to appear).
Preprint available for download from website of Greg W. Anderson.
.

Invariants for the elliptic normal quintic.
Preprint available for download from website of Tom Fisher.
.