On Tate-Shafarevich groups over galois extensions
On Tate-Shafarevich groups over galois extensions
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关于伽罗瓦扩展上的 Tate-Shafarevich 群
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
Hoseog Yu
中科院分区：
文献类型：
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作者：
Hoseog Yu
设A是定义在数域K上的阿贝尔簇。设L是K的有限Galois扩张，其Galois群为G，III（A/K）和III（A/L）分别表示A在K上和A在L上的Tate-Shafarevich群.假设这些群是有限的，我们计算[III（A/L）G]/[III（A/K）]和[III（A/K）]/[N（III（A/L））]，其中[X]是有限交换群X的阶。特别地，当L是K的二次扩张时，我们得到了一个简单的公式，它将[III（A/L）]，[III（A/K）]和[III（Ax/K）]联系起来，其中Ax是A的扭曲与G的非平凡特征标χ的乘积.
LetA be an abelian variety defined over a number fieldK. LetL be a finite Galois extension ofK with Galois groupG and let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups ofA overK and ofA overL. Assuming these groups are finite, we compute [III(A/L)G]/[III(A/K)] and [III(A/K)]/[N(III(A/L))], where [X] is the order of a finite abelian groupX. Especially, whenL is a quadratic extension ofK, we derive a simple formula relating [III(A/L)], [III(A/K)], and [III(Ax/K)] whereAx is the twist ofA by the non-trivial characterχ ofG.