Manin conjecture

In mathematics, the Manin conjecture describes the conjectural distribution of rational points on an algebraic variety relative to a suitable height function. It was proposed by Yuri I. Manin and his collaborators in 1989 when they initiated a program with the aim of describing the distribution of rational points on suitable algebraic varieties.

Their main conjecture is as follows. Let ${\displaystyle V}$ be a Fano variety defined over a number field ${\displaystyle K}$, let ${\displaystyle H}$ be a height function which is relative to the anticanonical divisor and assume that ${\displaystyle V(K)}$ is Zariski dense in ${\displaystyle V}$. Then there exists a non-empty Zariski open subset ${\displaystyle U\subset V}$ such that the counting function of ${\displaystyle K}$-rational points of bounded height, defined by

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