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Degree and the Brauer-Manin obstruction. (English) Zbl 1408.14086
Let \(X\) be a smooth, projective, geometrically integral variety over a number field \(k\). It is well-known that in general the Hasse principle may fail for \(X\) so that \(X(k)\) is empty while \(X(\mathbb{A}_k)\) is not. Sometimes (but not always – see [A. N. Skorobogatov, Invent. Math. 135, No. 2, 399–424 (1999; Zbl 0951.14013)] for a bielliptic counterexample) this failure can be explained by the Brauer-Manin obstruction. More precisely, there is a pairing \[ X(\mathbb{A}_k)\times \mathrm{Br}(X)\to \mathbb{Q}/\mathbb{Z} \] between the adelic points of \(X\) and its Brauer group such that for any subgroup \(B\subset\mathrm{Br}(X)\), the rational points \(X(k)\) are contained in the set \(X(\mathbb{A}_k)^B\) of adelic points which are orthogonal to all elements of \(B\).
Recent literature deals with the question how the torsion order of a Brauer class relates to the obstruction it can give. For a positive integer \(d\), denote by \(B[d^\infty]\) the \(d\)-primary part of \(B\) and by \(B[d^\perp]\) the prime-to-\(d\) part.
Say that the property \(\mathrm{BM}_{B,d}\) is satisfied for \(X\) if \(B[d^\infty]\) captures any Brauer-Manin obstruction to the Hasse principle, i.e. the following implication holds: \[ X(\mathbb{A}_k)^B=\emptyset \Rightarrow X(\mathbb{A}_k)^{B[d^\infty]}=\emptyset. \] Say that \(\mathrm{BM}_{B,d}^\perp\) is satisfied for \(X\) if \(B[d^\perp]\) never obstructs the Hasse principle, i.e. if the following implication holds: \[ X(\mathbb{A}_k)\neq\emptyset \Rightarrow X(\mathbb{A}_k)^{B[d^\perp]}\neq\emptyset. \] We omit the index \(B\) in the notation when \(B=\mathrm{Br}(X)\).
It has been shown that \(\mathrm{BM}_2^\perp\) holds for diagonal quartic surfaces over \(\mathbb{Q}\) [E. Ieronymou and A. N. Skorobogatov, Adv. Math. 270, 181–205 (2015; Zbl 1388.14071), E. Ieronymou and A. N. Skorobogatov, Adv. Math. 307, 1372–1377 (2017; Zbl 1388.14072)] (with a generalisation to number fields in [M. Nakahara, Adv. Math. 348, 512–522 (2019)]), Kummer varieties [A. N. Skorobogatov and Y. G. Zarhin, Pure Appl. Math. Q. 13, No. 2, 337–368 (2017; Zbl 1398.14029)] and del Pezzo surfaces of degree \(2\) [M. Nakahara, Adv. Math. 348, 512–522 (2019)]. On the other hand, in [P. Corn and M. Nakahara, Res. Number Theory 4, No. 3, Art. 33 (2018)] resp. [J. Berg and A. Várilly-Alvarado, “Odd order obstructions to the Hasse principle on general K3 surfaces”, arXiv:1808.00879] an obstruction by an algebraic resp. transcendental \(3\)-torsion Brauer class on a degree \(2\) K3 surface was constructed.
The present article relates the above properties to the degree of a variety. The authors ask whether degrees capture the Brauer-Manin obstruction on \(X\), which is defined to mean that \(\mathrm{BM}_d\) holds for any \(d\) which is the degree of a globally generated ample line bundle on \(X\). Apart from being a natural question, an answer may be relevant for algorithmic purposes.
The first main result is that torsors under abelian varieties satisfy \(\mathrm{BM}_{B,d}\) for any subgroup \(B\) and any \(d\) divisible by the period of the variety, in particular for any \(d\) which is the degree of a globally generated ample line bundle on \(X\). Taking \(B=\mathrm{Br}(X)[d^\perp]\), this shows that any prime-to-\(d\) Brauer class cannot obstruct. The proof relies on the observation that for any class \(\mathcal{A}\) in \(B[d^\perp]\), there exists an étale automorphism of \(X\) which annihilates \(\mathcal{A}\). For \(B=\mathrm{Br}(X)\) and semi-abelian varieties, one could also prove the theorem using Manin’s work under the assumption of finiteness of the Tate-Shafarevich group.
The authors then proceed to consider quotients of torsors under abelian varieties and prove that Kummer varieties satisfy \(\mathrm{BM}_2\). In combination with the aforementioned result by Skorobogatov-Zarhin, the appendix strengthens this to \(\mathrm{BM}_{B,2}\) for any subgroup \(B\). It is also shown that assuming the finiteness of the Tate-Shafarevich group, degrees capture the Brauer-Manin obstruction on bielliptic surfaces.
The last section proves that degrees capture the Brauer-Manin obstruction on minimal rational conic bundles and Severi-Brauer bundles over elliptic curves with finite Tate-Shafarevich group, which combined with previous results on del Pezzo surfaces implies an affirmative answer for all geometrically rational minimal surfaces. Finally a counterexample of a conic bundle over an elliptic curve, where degrees do not capture the Brauer-Manin obstruction, is given.
14G05 Rational points
11G35 Varieties over global fields
14F22 Brauer groups of schemes
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