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>