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关于阿贝尔簇的Tate-Shafarevich群

On Tate-Shafarevich groups of abelian varieties

Cristian D. Gonzalez-Avilés

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摘要：

设$K/F$是数域的有限伽罗瓦扩张，其伽罗瓦群为$G$，$A$是定义在$F$上的阿贝尔簇，${\cyr W}(A_{^{/ K}})$和${\cyr W}(A_{^{/ F}})$分别表示$A$在$K$上和$A$在$F$上的Tate-Shafarevich群.假设这些群是有限的，在对$A$和$K/F$的某些限制下，我们导出了$G$ -不变元的${\cyr W}(A_{^{/ K}})$的子群的阶的公式.作为推论，当$K/F$是二次扩张，$A^χ$是$A$的扭曲，由$G$的非平凡特征元$χ$，我们得到了一个简单的公式，它关系到${\cyr W}(A_{^{/ K}})$，${\cyr W}(A_{^{/ F}})$和${\cyr W}(A_{^{/ F}}^χ)$的阶.

英文摘要：

Let $K/F$ be a finite Galois extension of number fields with Galois group $G$, let $A$ be an abelian variety defined over $F$, and let ${\cyr W}(A_{^{/ K}})$ and ${\cyr W}(A_{^{/ F}})$ denote, respectively, the Tate-Shafarevich groups of $A$ over $K$ and of $A$ over $F$. Assuming that these groups are finite, we derive, under certain restrictions on $A$ and $K/F$, a formula for the order of the subgroup of ${\cyr W}(A_{^{/ K}})$ of $G$-invariant elements. As a corollary, we obtain a simple formula relating the orders of ${\cyr W}(A_{^{/ K}})$, ${\cyr W}(A_{^{/ F}})$ and ${\cyr W}(A_{^{/ F}}^χ)$ when $K/F$ is a quadratic extension and $A^χ$ is the twist of $A$ by the non-trivial character $χ$ of $G$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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