Let $f : X \to Y$ be a proper, dominant morphism of smooth varieties over a number field $k$. When is it true that for almost all places $v$ of $k$, the fibre $X_P$ over any point $P \in Y (k_v)$ contains a zero-cycle of degree $1$? We develop a necessary and sufficient condition to answer this question. The proof extends logarithmic geometry tools that have recently been developed by Denef and Loughran–Skorobogatov–Smeets to deal with analogous Ax–Kochen type statements for rational points.

中文翻译：

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零周期的算术相射性

令$ f：X \ to Y $是在数字场$ k $上的光滑变体的适当的，占优的态素。在几乎所有地方$ v $到$ k $的情况下，Y（k_v）$上任意点$ P \ x上的光纤$ X_P $包含度数$ 1 $的零周期是什么时候成立？我们为回答这个问题制定了必要的充分条件。该证明扩展了Denef和Loughran–Skorobogatov–Smeets最近开发的对数几何工具，用于处理有理点的类似Ax–Kochen类型陈述。