Abstract
Let mathbb R(C) be the function field of a smooth, irreducible projective curve over mathbb R. Let X be a smooth, projective, geometrically irreducible variety equipped with a dominant morphism f onto a smooth projective rational variety with a smooth generic fibre over mathbb R(C). Assume that the cohomological obstruction introduced by Colliot-Thélène is the only one to the local-global principle for rational points for the smooth fibres of f over mathbb R(C)-valued points. Then we show that the same holds for X, too, by adopting the fibration method similarly to Harpaz–Wittenberg.
Highlights
Let C be a smooth, geometrically irreducible projective curve over R
Let R(C) denote the function field of C, and for every x ∈ C(R) let R(C)x be the completion of R(C) with respect to the valuation furnished by x
We say that V satisfies the local-global principle for rational points if for every X in V the following holds: X (R(C)x ) = ∅ implies that X (R(C)) = ∅
Summary
Let C be a smooth, geometrically irreducible projective curve over R. There is an even more general result due to Scheiderer: Theorem 1.2 (Scheiderer) The local-global principle for rational points holds for smooth compactifications of homogeneous spaces over connected linear groups over R(C ). The failure of the local-global principle for this X was explained using a simple obstruction, analogous to the Brauer–Manin obstruction in the number field case, by Colliot-Thélène in [3] around 20 years ago, using unramified cohomology groups The result implies that the C T obstruction is the only one to the local-global principle for rational points for X when the generic fibre of f is the smooth compactification of a homogeneous space by a connected linear algebraic group by Theorem 1.2. We use arguments inspired by the work of Harpaz–Wittenberg in [11] to prove Theorem 5.3 in the sixth section, using the approximation theorem from the fourth section as a crucial ingredient
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