论文标题
Abelian品种的Shafarevich-Tate群体
Shafarevich-Tate groups of abelian varieties
论文作者
论文摘要
Shafarevich-Tate组$ W（\ Mathscr {a}）$衡量了Hasse原理的Abelian品种$ \ Mathscr {a} $的失败。使用Abelian品种与较高尺寸的非共同托里之间的对应关系，我们证明$ W（\ Mathscr {a}）\ cong cl〜（λ）\ oplus cl〜（λ）$或$ W（λ）$或$ w（\ mathscr {A}） \ oplus cl_ {〜\ MathBf {odd}}〜（λ）\ oplus cl_ {〜\ m mathbf {odd}}〜（λ）$，其中$ cl〜（λ）$是与非公认tori and $ 2^k $ cl divide的理想类$λ$相关的RING $λ$组的组组。详细考虑了具有复杂乘法的椭圆曲线的情况。
The Shafarevich-Tate group $W (\mathscr{A})$ measures the failure of the Hasse principle for an abelian variety $\mathscr{A}$. Using a correspondence between the abelian varieties and the higher dimensional non-commutative tori, we prove that $W (\mathscr{A})\cong Cl~(Λ)\oplus Cl~(Λ)$ or $W (\mathscr{A})\cong \left(\mathbf{Z}/2^k\mathbf{Z}\right) \oplus Cl_{~\mathbf{odd}}~(Λ)\oplus Cl_{~\mathbf{odd}}~(Λ)$, where $Cl~(Λ)$ is the ideal class group of a ring $Λ$ associated to the K-theory of the non-commutative tori and $2^k $ divides the order of $Cl~(Λ)$. The case of elliptic curves with complex multiplication is considered in detail.
