Nonisomorphic curves that become isomorphic over extensions of coprime degrees

TL;DR

This paper constructs examples of nonisomorphic algebraic curves over a field that become isomorphic over finite extensions of coprime degrees, revealing intricate behaviors of curves under field extensions and employing Galois cohomology techniques.

Contribution

It demonstrates the existence of nonisomorphic curves becoming isomorphic over coprime degree extensions and analyzes their genus and field characteristics, including explicit genus-2 examples.

Findings

Nonisomorphic curves can become isomorphic over coprime degree extensions.

Genus-0 curves cannot become isomorphic over such extensions.

Genus-1 examples exist only under specific conditions in finite fields.

Abstract

We show that one can find two nonisomorphic curves over a field K that become isomorphic to one another over two finite 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 and s be integers greater than 1 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 nonisomorphic 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 nonisomorphic genus-0 curves over an arbitrary field remain nonisomorphic over every odd-degree extension of the base field.