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

DOI:
https://doi.org/10.1090/S0002-9939-00-05887-1

Published electronically:
November 21, 2000

MathSciNet review:
1814093

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Abstract | References | Similar Articles | Additional Information

We show how for every integer one can explicitly construct distinct plane quartics and one hyperelliptic curve over 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 whose minimal polynomials over 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 , 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

Email:
however@alumni.caltech.edu

DOI:
https://doi.org/10.1090/S0002-9939-00-05887-1

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