### Plane quartics with Jacobians isomorphic to a hyperelliptic Jacobian, by Everett W. Howe

We show how for every integer *n* one can explicitly construct *n*
distinct plane quartics and one hyperelliptic curve over **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 **R** whose minimal
polynomials over **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
**C**, each with a pair of commuting involutions, are isomorphic to one
another.

Everett W. Howe <however@alumni.caltech.edu>