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edited May 3 2012 at 19:19

Wanderer
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Let $X$ be as smooth variety over a field $k$ of characteristic $0$.

Consider the following statements:

The variety $X$ has no $k((t))$-rational points.
No smooth compactification of $X$ has a $k$-rational point.

Are they these equivalent? If not, what additional assumptions on $X$ would make them equivalent? I'm particularly interested in the case where $X$ is a homogenous space of a "nice" algebraic group over $k$.

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asked May 3 2012 at 0:18

Wanderer
2,969●12●28

Rational points on smooth compactifications

Let $X$ be as smooth variety over a field $k$ of characteristic $0$.

Consider the following statements:

The variety $X$ has no $k((t))$-rational points.
No smooth compactification of $X$ has a $k$-rational point.

Are they equivalent? If not, what additional assumptions on $X$ would make them equivalent? I'm particularly interested in the case where $X$ is a homogenous space of a "nice" algebraic group over $k$.