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 describing the distribution of rational points on suitable algebraic varieties. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementFor a Fano variety V over a number field K with Zariski dense rational points and a height function H relative to the anticanonical divisor, there is a non-empty Zariski open subset U of V such that the number of K-rational points of U of height at most B is asymptotic to c B (log B)^(rho - 1) as B tends to infinity, where rho is the rank of the Picard group of V and c is a positive constant. 1 Proposed Year Progress Toward ResolutionProved for special families of varieties but still open in general; the constant c later received a conjectural interpretation by Peyre. 1 Classification
Resolution Status Prize Status
Prize Status (category) Sources
1. Manin Conjecture (Wikipedia)
Wikimedia FoundationLead section
It was proposed by Yuri I. Manin and his collaborators in 1989
Lead section, opening sentence
In mathematics, the Manin conjecture describes the conjectural distribution of rational points on an algebraic variety relative to a suitable height function.
Conjecture section, closing sentences
but is still open in general.
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