Rational points of bounded height on Fano varieties
type: Item
instance of: scholarly article
title: Rational points of bounded height on Fano varieties
zbMATH Open document ID: 0674.14012
DOI: 10.1007/BF01393904
author: Jens Franke
author: Yuri Tschinkel
author: Yuri I. Manin
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143659
review text: 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.
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 11G50
zbMATH DE Number: 4103229
zbMATH Keywords: Fano varieties
zbMATH Keywords: number of rational points
zbMATH Keywords: flag manifolds
zbMATH Keywords: height
zbMATH Keywords: Langlands-Eisenstein series
reviewed by: Daniel Bertrand
Wikidata QID: Q101496453
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4103229