论文标题
在Hasse原理上，圆锥形束超过程度扩展
On the Hasse Principle for conic bundles over even degree extensions
论文作者
论文摘要
令$ k $为一个数字字段，让$π\ colon x \ rightarrow \ mathbb {p} _k^1 $为平滑的圆锥捆。我们表明，如果$ x/k $具有四个几何奇异纤维，则具有$ x（\ mathbb {a} _k）\ neq \ emberyset $或非平凡的brauer group，则$ x $可以满足任何均匀度的扩展$ l/k $。此外，对于任意$ x $，我们表明的是，以辛泽尔的假设为条件，$ x $满足了所有$ k $的二次扩展几乎有限的大量原则。我们通过显示Brauer-Manin障碍物消失，然后在Colliot-Thélène和Sansuc之后应用Colliot-Thélène的结果来证明这些结果。
Let $k$ be a number field and let $π\colon X \rightarrow \mathbb{P}_k^1$ be a smooth conic bundle. We show that if $X/k$ has four geometric singular fibers with $X(\mathbb{A}_k)\neq \emptyset$ or non-trivial Brauer group, then $X$ satisfies the Hasse principle over any even degree extension $L/k$. Furthermore for arbitrary $X$ we show that, conditional on Schinzel's hypothesis, $X$ satisfies the Hasse principle over all but finitely many quadratic extensions of $k$. We prove these results by showing the Brauer-Manin obstruction vanishes and then apply fibration method results of Colliot-Thélène, following Colliot-Thélène and Sansuc.
