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Rational points on some pencils of conics with 6 singular fibres. (English) Zbl 0976.14014

The author writes: “Let \({\mathcal Y}\to\mathbb{P}^1\) be a pencil of conics defined over an algebraic number field \(k\). It is conjectured that the only obstruction to the Hasse principle on \({\mathcal Y}\), and also to weak approximation, is the Brauer-Manin obstruction; and it was shown by J.-L. Colliot-Thélène and Sir P. Swinnerton-Dyer [J. Reine Angew. Math. 453, 49-112 (1994; Zbl 0805.14010)] that this follows from Schinzel’s hypothesis.\(\dots\) If one does not assume Schinzel’s hypothesis, little is known. The only promising-looking line of attack is through the geometry of the universal torseurs on \({\mathcal Y}\); and these are much easier to study when \({\mathcal Y}\) has the special form \(U^2-cV^2=P(W)\), where \(c\) is a non-square in \(k\) and \(P(W)\) is a separable polynomial in \(k[W]\). By writing \(W=X/Y\) we can take the solubility of this equation into the equivalent (though ungeometric) problem of the solubility of \(U^2-cV^2=f(X,Y)\) in \(k\), where \(f\) is homogeneous of even degree; here \(\deg f\) is \(1+\deg P\) or \(\deg P\). The simplest non-trivial case is that of Châtelet surfaces, when \(P(W)\) has degree 3 or 4; in this case the conjecture was proved by J.-L. Colliot-Thélène, J.-J. Sansuc, and P. Swinnerton-Dyer [J. Reine Angew. Math. 373, 37-107 (1987; Zbl 0622.14029)] and 374, 71-168 (1987; Zbl 0622.14030)]. The object of this paper is to prove the conjecture when \(\deg f=6\) and \(f=f_4f_2\) over \(k\), where \(\deg f_4=4\) and \(\deg f_2=2\).”
The author solves this problem by explicitly constructing a finite set \(S\) of projective algebraic varieties defined over \(k\), with the following properties:
(i) if \(V\in S\), then \(V(k)\neq \emptyset\) as soon as \(V(k_v)\neq \emptyset\) for every completion \(k_v\) of \(k\);
(ii) if \(V(k)\neq \emptyset\) for some \(V\) in \(S\), then the variety \(W:U^2-cV^2 =f(X,Y)\) contains a Zariski dense set of points defined over \(k\);
(iii) if the equation \(U^2-cV^2= f(X,Y)\) has a non-trivial solution in \(k\), then \(V(k)\neq\emptyset\) for some \(V\) in \(S\).
Moreover, each of the varieties in \(S\) can be proved to be isomorphic to a factor of an universal torsor associated to the corresponding pencil of conics \({\mathcal Y}:U^2-cV^2=P(W)\).
Reviewer: B.Z.Moroz (Bonn)

MSC:

14G05 Rational points
14C21 Pencils, nets, webs in algebraic geometry
11D72 Diophantine equations in many variables
11G35 Varieties over global fields
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References:

[1] Colliot-Thélène, J.-L.and Sansuc, J.-J., La descente sur les variétés rationnelles II, Duke Math. J.54 (1987), 375-492. · Zbl 0659.14028
[2] Colliot-Thélène, J.-L., Sansuc, J.-J. and Swinnerton-Dyer, Sir Peter, Intersections of two quadrics and Châtelet surfaces, J. reine angew. Math.373 (1987), 37-107 and 374 (1987), 72-168. · Zbl 0622.14030
[3] Colliot-Thélène, J.-L. and Swinnerton-Dyer, Sir Peter, Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties, J. reine angew. Math.453 (1994), 49-112. · Zbl 0805.14010
[4] Manin, Y.I., Cubic Forms, algebra, geometry, arithmetic. (2nd edition, North-Holland, 1986) · Zbl 0582.14010
[5] Swinnerton-Dyer, Sir Peter, The Brauer group of cubic surfaces, Math. Proc. Camb. Phil. Soc.113 (1993), 449-460. · Zbl 0804.14018
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