Let $X$ be a twist of the $n$-th projective space, seen as a $K$-variety for some number field $K$. For $n = 1$, the Hasse principle holds for $X$.
My question is: for which $n >1$ does the Hasse principle also hold?
Thank you.
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3$\begingroup$ I don't understand the question, Projective space has a rational point, so the Hasse principle trivially holds. Are you interested in whether the Hasse principle holds for Brauer-Severi varieties (i.e. twists of projective space)? Here the Hasse principle is classical and can be proved using class field theory. $\endgroup$– Daniel LoughranCommented Sep 10, 2014 at 7:32
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$\begingroup$ yes sorry I forgot to write twists $\endgroup$– user57473Commented Sep 10, 2014 at 7:47
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1 Answer
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As Daniel mentioned, the answer is yes. This is Theorem 4.5.11 in Bjorn Poonen's Rational Points on Varieties.
