论文标题
在Brauer-Manin配对的内核上
On the kernel of the Brauer-Manin pairing
论文作者
论文摘要
令$ \ Mathcal X $为常规方案，在$ p $ - adic字段的整数上均固定，并具有通用光纤$ x $和特殊光纤$ \ Mathcal x_s $。我们研究Brauer-Manin配对$ BR（X）\ Times CH_0（X）\ to \ Mathbb Q/\ Mathbb Z $的左核$ br（\ Mathcal x）$。我们的主要结果是还原图的内核$ br（\ Mathcal x）\ to Br（\ Mathcal x_s）$是$（\ Mathbb Q/\ Mathbb z [\ frac {1} {1} {p} {p} {p} {p} {p}]）^s \ oplus（\ mathbb q/prite $ s）$ priite $ s）+s $ s+s+s+s+s+s+s+s+s+s+ ρ_{\ Mathcal X_S}-ρ_x-i+1 $，对于$ρ_{\ Mathcal x_s} $和$ρ_x$ $ \ MATHCAL x_s $和$ x $的PICARD数字，以及$ i $ $ i $ $ i $ $ $ i $ $ $ i $ $ \ mathcal x_s x_s $的不可修复组件。此外，我们表明$ t> 0 $表示$ s> 0 $。
Let $\mathcal X$ be a regular scheme, flat and proper over the ring of integers of a $p$-adic field, with generic fiber $X$ and special fiber $\mathcal X_s$. We study the left kernel $Br(\mathcal X)$ of the Brauer-Manin pairing $Br(X)\times CH_0(X)\to \mathbb Q/\mathbb Z$. Our main result is that the kernel of the reduction map $Br(\mathcal X)\to Br(\mathcal X_s)$ is the direct sum of $(\mathbb Q/\mathbb Z[\frac{1}{p}])^s\oplus (\mathbb Q/\mathbb Z)^t$ and a finite $p$-group, where $s+t= ρ_{\mathcal X_s}-ρ_X-I+1$, for $ρ_{\mathcal X_s}$ and $ρ_X$ the Picard numbers of $\mathcal X_s$ and $X$, and $I$ the number of irreducible components of $\mathcal X_s$. Moreover, we show that $t>0$ implies $s>0$.
