Manin conjecture

The Manin conjecture predicts the asymptotic number of rational points of bounded height on Fano varieties over number fields, with a leading constant and growth rate determined by geometric and arithmetic data.

Height function: A function measuring the arithmetic complexity of rational points, often through projective coordinates. The conjecture counts points below a height bound.
Fano variety: A projective variety whose anticanonical divisor is ample. These are the varieties whose rational points the conjecture predicts counting.
Yuri Manin: A Russian mathematician known for work in algebraic geometry, number theory, and mathematical physics. He proposed the conjectural framework for counting rational points on Fano varieties.
Projective space: A variety whose points parametrize lines through the origin in a vector space. Its rational points admit a classical height count matching the predicted growth.
Accumulating subvariety: A subvariety whose rational points contribute disproportionately to the count on an ambient variety. Such subsets can dominate the count unless the conjectural formula excludes them.
Peyre's constant: A product of geometric and local arithmetic factors proposed as the leading constant in Manin's asymptotic formula. It refines the predicted count beyond its power and logarithmic growth.
Rational point: A point on an algebraic variety whose coordinates lie in a specified field. The conjecture counts points defined over the number field.
Vasily Iskovskikh: A mathematician whose work helped establish the birational geometry of algebraic surfaces and higher-dimensional varieties. His work on del Pezzo surfaces underpinned early examples motivating the conjecture.
Toric variety: An algebraic variety containing an algebraic torus as a dense open subset, with the torus action extending to the variety. Toric methods prove the predicted asymptotics for broad families of examples.
Brauer–Manin obstruction: An obstruction to rational points or weak approximation arising from the Brauer group and adelic conditions. It helps explain when local conditions suppress points expected by naive geometric estimates.

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Birch's theoremRelated: It shares the goal of predicting arithmetic point counts, but addresses broader geometric settings.