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Plane quartics with Jacobians isomorphic to a hyperelliptic Jacobian

Author: Everett W. Howe
Journal: Proc. Amer. Math. Soc. 129 (2001), 1647-1657
MSC (2000): Primary 14H40; Secondary 14H45
Published electronically: November 21, 2000
MathSciNet review: 1814093
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We show how for every integer $n$ one can explicitly construct $n$ distinct plane quartics and one hyperelliptic curve over ${\mathbf C}$ all of whose Jacobians are isomorphic to one another as abelian varieties without polarization. When we say that the curves can be constructed ``explicitly'', we mean that the coefficients of the defining equations of the curves are simple rational expressions in algebraic numbers in ${\mathbf R}$ whose minimal polynomials over ${\mathbf Q}$ can be given exactly and whose decimal approximations can be given to as many places as is necessary to distinguish them from their conjugates. We also prove a simply-stated theorem that allows one to decide whether or not two plane quartics over ${\mathbf C}$, each with a pair of commuting involutions, are isomorphic to one another.

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Additional Information

Everett W. Howe
Affiliation: Center for Communications Research, 4320 Westerra Court, San Diego, California 92121-1967

Keywords: Curve, Jacobian, polarization, Torelli, quartic
Received by editor(s): December 1, 1998
Received by editor(s) in revised form: October 9, 1999
Published electronically: November 21, 2000
Communicated by: Ron Donagi
Article copyright: © Copyright 2000 American Mathematical Society

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