(Both the published version and the arXiv version are available online.)
We show that one can find two non-isomorphic curves over a field K that become isomorphic over two extensions of K whose degrees over K are coprime to one another.
More precisely, let K0 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 K0, 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 then C and D can have genus 1 if and only if {r,s} = {2,3} and K is an odd-degree extension of F3. 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.
Michael Zieve: home page publication list