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We show that one can find two nonisomorphic curves over a field K that becomeisomorphic to one another over two finite extensions of K whose degrees over Kare coprime to one another.  More specifically, let K_0 be an arbitrary prime field and let r and s beintegers greater than 1 that are coprime to one another. We show that one canfind a finite extension K of K_0, a degree-r extension L of K, a degree-sextension M of K, and two curves C and D over K such that C and D becomeisomorphic to one another over L and over M, but not over any propersubextensions of L/K or M/K.  We show that such C and D can never have genus 0, and that if K is finite, Cand D can have genus 1 if and only if {r,s} = {2,3} and K is an odd-degreeextension of F_3. On the other hand, when {r,s}={2,3} we show that genus-2examples occur in every characteristic other than 3.  Our detailed analysis of the case {r,s} = {2,3} shows that over every finitefield K there exist nonisomorphic curves C and D that become isomorphic to oneanother over the quadratic and cubic extensions of K.  Most of our proofs rely on Galois cohomology. Without using Galoiscohomology, we show that two nonisomorphic genus-0 curves over an arbitraryfield remain nonisomorphic over every odd-degree extension of the base field.

Nonisomorphic curves that become isomorphic over extensions of coprime degrees