Math Conjectures

Thin-Set Manin–Peyre Conjecture

OPENLandmarkConjectureProposed 1989–1995 · Standard version

Canonical statement

Let XX be a smooth projective geometrically integral Fano variety over a number field FF, assume that X(F)X(F) is not a thin subset of XX, and let HH be an adelically metrized anticanonical height. Put ρ=rank⁡Pic⁡(X)\rho=\operatorname{rank}\operatorname{Pic}(X). There exists a thin subset Z⊂X(F)Z\subset X(F), meaning a finite union of subsets contained in proper closed subvarieties and images of rational points under generically finite dominant maps of degree greater than one, such that
#{x∈X(F)∖Z:H(x)≤B}∼cX,HB(log⁡B)ρ−1 \#\{x\in X(F)\setminus Z:H(x)\le B\} \sim c_{X,H}B(\log B)^{\rho-1}
as B→∞B\to\infty, where cX,H=α(X)β(X)τH(X(AF)Br⁡)c_{X,H}=\alpha(X)\beta(X)\tau_H(X(\mathbb A_F)^{\operatorname{Br}}) is Peyre's constant.
View source LaTeX
Let \(X\) be a smooth projective geometrically integral Fano variety over a number field \(F\), assume that \(X(F)\) is not a thin subset of \(X\), and let \(H\) be an adelically metrized anticanonical height. Put \(\rho=\operatorname{rank}\operatorname{Pic}(X)\). There exists a thin subset \(Z\subset X(F)\), meaning a finite union of subsets contained in proper closed subvarieties and images of rational points under generically finite dominant maps of degree greater than one, such that
\[
  \#\{x\in X(F)\setminus Z:H(x)\le B\}
  \sim c_{X,H}B(\log B)^{\rho-1}
\]
as \(B\to\infty\), where \(c_{X,H}=\alpha(X)\beta(X)\tau_H(X(\mathbb A_F)^{\operatorname{Br}})\) is Peyre's constant.
AG-DIO-024:[FrankeManinTschinkel1989Heights][Peyre1995HeightsTamagawa][LehmannSenguptaTanimoto2022Manin]
Algebraic geometryNumber theory

Notes

Manin's conjecture predicts how rational points of bounded anticanonical height are distributed on a Fano variety. In the basic Fano case the count should grow like B(log⁡B)ρ−1B(\log B)^{\rho-1}, where ρ\rho is the rank of the rational Picard group [FrankeManinTschinkel1989Heights]. Peyre refined this prediction by defining the leading constant from the effective cone, Galois cohomology, and a Tamagawa measure on the Brauer–Manin set [Peyre1995HeightsTamagawa].

The asymptotic has been proved for many toric, flag, equivariant, and low-dimensional varieties. The original expectation that one could always delete only a proper Zariski-closed subset has counterexamples: exceptional families of rational points can arise as images of generically finite covers and must instead be removed as a thin set. Modern geometric work identifies and controls these accumulating contributions [LehmannSenguptaTanimoto2022Manin].

What remains is a proof for every smooth projective Fano variety whose rational points are non-thin, together with a canonical enough description of the exceptional thin set. The entry therefore records the modern thin-set Manin–Peyre version; it does not revive the false unqualified open-subset formulation.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.