(An official version and a preprint are available online.)

We show that one can find two non-isomorphic
curves over a field *K* that become isomorphic
to one another over two extensions of *K*
whose degrees over *K* are coprime to one another.

More specifically, let *K*_{0} be an arbitrary prime field and let
*r* > 1 and *s* > 1 be integers that are coprime
to one another. We show that one can find a
finite extension *K* of *K*_{0}, a degree-*r*
extension *L* of *K*, a degree-*s* extension
*M* of *K*, and two curves *C* and *D* over *K*
such that *C* and *D* become isomorphic to one
another over *L* and over *M*, but not over
any proper subextensions of *L/K* or *M/K*.

We show that such *C* and *D* can never have
genus 0, and that if *K* is finite, *C*
and *D* can have genus 1 if and only if
{*r,s*} = {2,3} and *K* is an odd-degree
extension of **F**_{3}. On the other hand, when
{*r,s*} = {2,3} we show that genus-2 examples
occur in every characteristic other than 3.

Our detailed analysis of the case {*r,s*} = {2,3}
shows that over every finite field *K*
there exist non-isomorphic curves *C* and *D*
that become isomorphic to one another over the
quadratic and cubic extensions of *K*.

Most of our proofs rely on Galois cohomology. Without using Galois cohomology, we show that two non-isomorphic genus-0 curves over an arbitrary field remain non-isomorphic over every odd-degree extension of the base field.