The fibration method over real function fields
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Mathematische Annalen
The fibration method over real function fields Ambrus Pál1 · Endre Szabó2 Received: 12 June 2019 / Revised: 17 July 2020 © The Author(s) 2020
Abstract Let R(C) be the function field of a smooth, irreducible projective curve over 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 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 R(C)-valued points. Then we show that the same holds for X , too, by adopting the fibration method similarly to Harpaz–Wittenberg. Mathematics Subject Classification 14G05 · 14P05
1 Introduction 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. Now let V be a class of geometrically irreducible projective varieties over R(C). 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)) = ∅.
x∈C(R)
There are classes of varieties when this local-global principle holds:
Communicated by Wei Zhang.
B
Ambrus Pál [email protected] Endre Szabó [email protected]
1
Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2AZ, UK
2
Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13-15, Budapest 1053, Hungary
123
A. Pál, E. Szabó
Theorem 1.1 (Witt) The local-global principle for rational points holds for smooth quadric hypersurfaces of dimension at least one over R(C). Proof See [15,16].
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). Proof See [14].
However similarly to varieties over number fields, there are some reasonably simple counter-examples to this local-global principle. The following counterexample is due to Racinet: let C = P1R and let U be the affine surface over R(C) = R(t) given by the following equation: a 2 + b2 = (c2 + t)(tc2 + c − 1) in the variables a, b and c, and let X be a smooth projective model of U . Then X (R(X )x ) is non-empty for every x ∈ C(R), but X (R(C)) is empty. 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. He constructed a subset ⎛
⎝
⎞C T X (R(C)x )⎠
⊆
x∈C(R)
X (R(C)x )
x∈C(R)
which contains X (R(C)). We will review this construction in the next section. Later Ducros has found a simple topological obstruction equivalent to this obstruction which we will describe next. By resolution of singularities the
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