AI 中文总结
研究有限域扩张下布劳尔 - 马宁障碍对有理点存在性的影响，针对数域上\((p, mp)\)-范数丛，证明基变换到特定有限扩张后障碍消失，对\((p,2p)\)-范数丛给出更简便条件，且证明可除性假设通常最优。
AI 中文摘要
我们研究了有限域扩张下布劳尔 - 马宁障碍对有理点存在性的影响。对于数域上的\((p, mp)\)-范数丛，我们证明在基变换到度数满足依赖于\(m\)的合适\(p\)可除性条件的有限扩张后，布劳尔 - 马宁障碍消失。进一步表明，对于\(p = 2\)或\(3\)的\((p,2p)\)-范数丛，只需假设扩张度数可被\(p\)整除。还通过构造一个二次扩张上布劳尔 - 马宁障碍仍然存在的圆锥丛证明了可除性假设通常是最优的。
英文摘要
We study the behaviour of the Brauer--Manin obstruction to the existence of rational points under finite field extensions. For $(p, mp)$-normic bundles over number fields, we prove that the Brauer--Manin obstruction vanishes after base change to finite extensions whose degrees satisfy suitable $p$-divisibility conditions depending on $m$. We further show that, for $(p,2p)$-normic bundles with $p=2$ or $3$, it is enough to assume that the extension degree is divisible by $p$. We also prove that the divisibility hypothesis is, in general optimal, by constructing a conic bundle for which the Brauer--Manin obstruction persists over a quadratic extension.
