Rational points of bounded height on Fano varieties. (English) Zbl 0674.14012

Franke, Jens; Manin, Yuri I.; Tschinkel, Yuri

Invent. Math. 95, No. 2, 421-435 (1989).

In this beautiful written paper, the authors study the following conjecture of Yu. I. Manin: if \(V\) is an algebraic variety whose canonical bundle is the inverse of an ample one, say \(D\), the number of \(F\)-rational points on \(V\) with (exponential) \(D\text{-height}\leq H\) should grow like a constant multiple of \(H(\log(H))^{\text{rk}(\text{Pic}(V))-1}\), as soon as \(F\) is sufficiently large and \(V\) satisfies certain non-degeneracy assumptions. The conjecture is shown to be stable (in a strong form) under products and consistent (in a weak sense) with results from the circle method when \(V\) is a complete intersection. The greater part of the paper is concerned with generalized flag manifolds, where the conjecture is proved through an identification of the “zeta function” associated to the \(D\)-height with a Langlands-Eisenstein series.

Reviewer: Daniel Bertrand (Paris)

MSC:

14G05	Rational points
14J45	Fano varieties
14G25	Global ground fields in algebraic geometry
11G50	Heights

Keywords:

Fano varieties; number of rational points; flag manifolds; height; Langlands-Eisenstein series

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