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Very strong approximation for certain algebraic varieties. (English) Zbl 1339.14018

The paper under review deals with strong approximation for rational points of subvarieties of a torus over function fields, or over \(\mathbb Q\) and an imaginary quadratic field if the torus is split. The main result has application towards a conjecture of Harari-Voloch and to adelic points on subvarieties of tori.
Given an algebraic variety \(X\) defined over a number field \(k\), every time the set \(X(k)\) is non empty this implies that all the sets \(X(k_v)\) are also non empty, where \(k_v\) denotes the completion of \(k\) with respect to any place \(v\). An interesting and very deep question in Arithmetic Geometry is to describe conditions that guarantees that the converse holds, namely if we have a point in \(X(k_v)\) for every place \(v\) of \(k\) then there exists a \(k\)-point in \(X\). The famous Hasse-Minkowski Theorem asserts that this holds whenever \(X\) is a quadric defined over a number field. Since then the property has been known as the Hasse Principle: an algebraic variety over a number field \(k\) has a \(k\) point if and only if it has a \(k_v\) point for every completion of \(k\) at a place \(v\). It is known that there are counterexamples to the Hasse principle, already for cubic forms.
Yu. I. Manin [in: Actes Congr. internat. Math. 1970, No.1, 401–411 (1971; Zbl 0239.14010)] introduced an obstruction the the Hasse principle, the Brauer-Manin obstruction, that explains most of the examples of the failure of the Hasse principle. However this has been proved to be insufficient to explain all of the failures, as showed by A. N. Skorobogatov [Invent. Math. 135, No. 2, 399–424 (1999; Zbl 0951.14013)].
Formally, given a variety \(X\) defined over a number field \(k\), if \(\mathbb A_k\) denote the adelic ring of \(k\), one has that \[ X(k) \subset X(\mathbb A_k). \] The fundamental observation of Manin was that there is a natural pairing between the adelic points of \(X\) and the Brauer Group of \(X\), where the Brauer Group is defined as \[ \mathrm{Br}(X):= H^2_{\text{ét}}(X,\mathbb G_m), \] defined by \[ X(\mathbb A_k)) \times\mathrm{Br}(X)\to\mathbb Q/\mathbb Z. \] Every rational point \(x\in X(k)\) lies in the left kernel of the pairing map. Therefore, if we denote by \(X(\mathbb A_k)^{\mathrm{Br}(X)}\) the left kernel, we have that \[ X(k) \subset X(\mathbb A_k)^{\mathrm{Br}(X)}\subset X(\mathbb A_k). \] In particular if the left kernel of the pairing is trivial, then there cannot exists any \(k\)-rational point in \(X\).
In [J.-L. Colliot-Thélène and F. Xu, Compos. Math. 145, No. 2, 309–363 (2009; Zbl 1190.11036)] this setting was extended to the study of integral points. Given a finite set of places \(S\) of \(k\) one defines the \(S\)-adeles to be the projection of \(\mathbb A_k\) to \(\prod_{v\notin S}k_v\). This comes equipped with a natural projection map \[ P_S: X(\mathbb A_k) \to X(\mathbb A_k^S) \] Then the strong approximation property with Brauer-Manin obstruction off \(S\) holds for \(X\) (SAP-BM off S for short) if \(X(k)\) is dense in \(P_S(X(\mathbb A_k)^{\mathrm{Br}(X)})\) under the diagonal map (with adelic topology). The {it very} strong approximation property (VSAP-BM in short) holds whether in addition \[ X(k)=P_S(X(\mathbb A_k)^{\mathrm{Br}(X)}). \] The author show that SAP-BM off \(S\) is equivalent to the fact that the Brauer-Manin obstruction is the only obstruction for the existence of integral points for every model of \(X\) over the ring of \(S\)-integers (Theorem 2.10).
The main results of the paper concert VSAP-BM for subvarieties of a Torus over a global function field \(F\) (or for \(k=\mathbb Q\) or an imaginary quadratic field when the Torus is split) – Theorem 1.2, and a generalization to arbitrary algebraic varieties that reads as follows: Theorem 1.3. Any algebraic variety over \(\mathbb Q\) or an imaginary quadratic field, or a global function field, always contains a dense open subvariety which satisfies VSAP-BM off \(S\) for any finite subset of places \(S\) containing the archimedean ones. As an application the authors prove Harari-Voloch Conjecture, introduced in [D. Harari and J. F. Voloch, Math. Proc. Camb. Philos. Soc. 149, No. 3, 413–421 (2010; Zbl 1280.11038)], for hyperbolic affine curves defined over a global function field. At the end of the paper the author prove a result on rational points on finite subscheme of a torus, analogue to Theorem 3.11 of M. Stoll [Algebra Number Theory 1, No. 4, 349–391 (2007; Zbl 1167.11024].

MSC:

14G05 Rational points
14G25 Global ground fields in algebraic geometry
11G35 Varieties over global fields
14F22 Brauer groups of schemes
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References:

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