Jacobians of genus one curves

Journal of Number Theory Cover 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.

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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 | 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


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