摘要翻译：
我们证明了在域K上可以找到两条非同构曲线，这两条曲线在K的两个有限扩张上是同构的，它们在K上的度是互质的。更具体地说，设K0是任意素数域，设r和s是大于1的整数，它们是互质的。我们证明了可以找到K0的有限扩张K,K的次R扩张L,K的次S扩张M,以及K上的两条曲线C和D，使得C和D在L和M上同构，但不能在L/K或M/K的任何适当子扩张上同构。我们证明了这样的C和D永远不可能有亏格0，如果K是有限的，则C和D可以有亏格1，当且仅当{r,s}={2,3}且K是F_3的奇次扩张。另一方面，当{r,s}={2,3}时，我们证明了在除3以外的每个特征中都有属2的例子。对{r,s}={2,3}的详细分析表明，在每个有限域K上，存在着在K的二次扩张和三次扩张上相互同构的非同构曲线C和D。我们的证明大多依赖于Galois上同调。在不使用Galois上同调的情况下，我们证明了任意域上的两个非同构genus-0曲线在基域的每个奇次扩张上保持非同构。
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英文标题：
《Nonisomorphic curves that become isomorphic over extensions of coprime
  degrees》
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作者：
Daniel Goldstein, Robert M. Guralnick, Everett W. Howe, Michael E.
  Zieve
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最新提交年份：
2008
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  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.
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PDF链接：
https://arxiv.org/pdf/0801.4614