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Nonisomorphic curves that become isomorphic over extensions of coprime degrees. (English) Zbl 1160.14020
Let $$C$$, $$D$$ be curves over a field $$K$$, that are twists of one another (they become isomorphic over some field extension). One says that an extension $$L$$ of $$K$$ is a minimal isomorphism extension for $$C$$ and $$D$$ if these curves become isomorphic over $$L$$, but not over any proper subextension of $$L/K$$. Let $$K_0$$ be a prime field and let $$r>1$$ and $$s>1$$ be two coprime integers. In this paper it is shown that there exist curves $$C$$ and $$D$$ over a finite extension $$K$$ of $$K_0$$ that are twists of one another and that have minimal isomorphism extensions of degrees $$r$$ and $$s$$ over $$K$$. The proof for the case $$K_0$$ finite uses Galois cohomology, and it is based on the existence of a curve $$X$$ with a concrete structure (and Galois action) of the geometric automorphism group. The proof for the case $$K_0=\mathbb{Q}$$ is based on explicit constructions.
In addition, the authors take a closer look at the possibilities when $$C$$ and $$D$$ have small genus. They show that $$C$$ and $$D$$ can never have genus $$0$$, and they can have genus $$1$$ only if $$\{r,s\}=\{2,3\}$$, the field $$K$$ is an odd degree extension of $$\mathbb{F}_3$$ and $$C$$, $$D$$ have supersingular Jacobians. On the other hand, when $$\{r,s\}=\{2,3\}$$, they show that genus-$$2$$ examples occur in every characteristic other than $$3$$, whereas for $$\{r,s\}\neq\{2,3\}$$ the genus-$$2$$ examples are very special.
The paper finishes with an open question: given two linearly disjoint finite extension fields $$L$$ and $$M$$ of a field $$K$$, do there exist curves $$C$$ and $$D$$ over $$K$$ having $$L$$ and $$M$$ as minimal isomorphism extensions?

MSC:
 14H45 Special algebraic curves and curves of low genus 14H25 Arithmetic ground fields for curves 14H37 Automorphisms of curves
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