Rational points of bounded height on Fano varieties
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.
scientific article; zbMATH DE number 4103229