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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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