Tamagawa measures on universal torsors and points of bounded height on Fano varieties. (English) Zbl 0959.14007

Salberger, Per

Peyre, Emmanuel (ed.), Nombre et répartition de points de hauteur bornée. Paris: Société Mathématique de France, Astérisque. 251, 91-258 (1998).

Let \(X\) be a projective algebraic variety defined over a number field \(k\). Assume that the set \(X(k)\) of rational points is Zariski-dense and fix a height function \(H\) on \(X\). Then one can try to count the number \(n_U (B)\) of rational points of height at most \(B\) in a dense open Zariski subset \(U \subset X\). More precisely, it is an important question to understand the asymptotic growth of \(n_U(B)\) when \(B\) goes to infinity. For example, results of this kind were obtained by Néron for abelian varieties and by Schanuel for the projective space.
Assume further that \(X\) is Fano, that is, the anticanonical sheaf \(-K_X\) is ample. In the paper under review, the author restricts to the case when \(-K_X\) is very ample, which makes no big difference. In this case it is possible to take for \(H\) the anticanonical height. According to Manin’s conjectures, it is expected that for a number of Fano varieties, the asymptotic behavior of \(n_U(B)\) looks like \[ n_U(B)\sim CB (\log B)^{r-1}\tag{1} \] where \(C\) is a constant, \(r\) is the rank of the Picard group of \(X\) and \(U\) is sufficiently small (because certain closed subvarieties can contain too many rational points, e.g. lines on cubic surfaces). Although (1) cannot hold for all Fano varieties V. V. Batyrev and Y. Tschinkel gave a counterexample [cf. C. R. Acad. Sci., Paris, Sér. I 323, No. 1, 41-46 (1996; Zbl 0879.14007)], it was established for several classes of such varieties. It is therefore interesting to find a common framework for these situations.
It was Peyre who was the first to give an interpretation of the constant \(C\) in the formula (1). This interpretation was refined by V. V. Batyrev and Y. Tschinkel [Int. Math. Res. Not. 1995, No. 12, 591-635 (1995; Zbl 0890.14008)]. In the paper under review, the author uses universal torsors \({\mathcal T}\) over \(X\) to obtain a better understanding of \(C\). More precisely, he relates \(C\) to the volumes of some adelic spaces defined by \({\mathcal T}\). In particular, a new kind of Tamagawa number is defined for \({\mathcal T}\) (whereas Peyre had introduced Tamagawa numbers for Fano varieties). The idea of using universal torsors for counting the rational points on Fano varieties (they had initially been introduced by J.-L. Colliot-Thélène and J.-J. Sansuc for the study of the Hasse principle and weak approximation on rational varieties [Duke Math. J. 54, 375-492 (1987; Zbl 0659.14028)] seems to be especially important in the recent developments of this field [see for example E. Peyre in: Nombre et répartition de points de hauteur bornée, Astérisque 251, 259-298 (1998) in the same collection as the paper under review].
Let us describe briefly the contents of the 11 sections of the paper. In sections 1 and 2, a theory of analytic manifolds over locally compact fields is developed. This is useful in dealing with the \(v\)-adic analytic manifold associated to a \(k_v\)-variety \(X_v\) (where \(k_v\) is a non-Archimedean completion of \(k)\), and in particular with measures associated to \(X_v\). In section 3, the author considers torsors \({\mathcal T}\) over a \(K\)-variety \(X\) under an algebraic \(K\)-group \(G\) (where \(K\) is a local field) and relative norms associated to them. In the case when \(G\) is a torus, there is a canonical relative norm (the order norm), which can be used (starting from a norm on the analytic manifold \(X(k_v))\) to define a norm and a measure on \({\mathcal T}(k_v)\). Section 4 is devoted to the adelic space \(X(\mathbb{A}_k)\); Peyre’s adelic metrics and measures are generalized (for example to relative situations). In section 5 Peyre’s Tamagawa measures and numbers are extended to universal torsors; this implies generalizing some of the results by Colliot-Thélène and Sansuc to torsors over a base scheme (and not only over a field). Manin’s reciprocity law (or “Brauer-Manin obstruction”) is reviewed in section 6. It is used to prove an important formula (theorem 6.19) relating the Tamagawa numbers defined by Peyre to the Tamagawa numbers defined in section 5 for universal torsors. In section 7 one restricts to the case when \(X\) is a Fano variety; Peyre’s Tamagawa conjecture on the constant \(C\) is refined by means of the Tamagawa numbers for universal torsors \({\mathcal T}\). The geometry of \({\mathcal T}\) for a toric variety \(X\) is described in section 8. When the ground field is local, Batyrev-Tschinkel’s norms on \(X\) induce norms on \({\mathcal T}\), which are described in section 9. Section 10 deals with toric varieties over a number field. The notion of canonical toric splittings (defined in the previous section) is used to define heights and to give an interpretation of the constant \(C\) in terms of volumes of adelic spaces associated to \({\mathcal T}\). In section 11, a new proof of the Manin-Peyre conjecture for split toric \(\mathbb{Q}\)-varieties [this result is due to V. V. Batyrev and V. Tschinkel, J. Algebr. Geom. 7, No. 1, 15-53 (1998; Zbl 0946.14009)] is given. The asymptotic formula is a bit more precise, and one may hope that this new method (using the machinery of universal torsors) can be applied to other classes of Fano varieties.
This impressive paper is an important contribution to the domain. It gives beautiful applications of the notion of universal torsor, and the reader can also use the paper to learn certain technical results in a very general context.
For the entire collection see [Zbl 0909.00033].

Reviewer: David Harari (Strasbourg)

MSC:

14G05	Rational points
14J45	Fano varieties
14G40	Arithmetic varieties and schemes; Arakelov theory; heights
14G25	Global ground fields in algebraic geometry
11G50	Heights

Keywords:

Brauer-Manin obstruction; number of rational points; height function; anticanonical height; universal torsors; Tamagawa number

Citations:

Zbl 0879.14007; Zbl 0890.14008; Zbl 0659.14028; Zbl 0946.14009