A cubic hypersurface in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {P}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> defined over <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation> is given by the vanishing locus of a cubic form f in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> variables. It is conjectured that when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n \ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, such cubic hypersurfaces satisfy the Hasse principle. This is now known to hold on average due to recent work of Browning, Le Boudec, and Sawin. Using this result, we determine the proportion of cubic hypersurfaces in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {P}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>, ordered by the height of f, with a rational point for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$n \ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> explicitly as a product over primes p of rational functions in p. In particular, this proportion is equal to 1 for cubic hypersurfaces in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {P}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n \ge 9$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>9</mn> </mrow> </math></EquationSource> </InlineEquation>; for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$100\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>100</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> of cubic hypersurfaces, this recovers a celebrated result of Heath-Brown that non-singular cubic forms in at least 10 variables have rational zeros. In the <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$n=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> case, we give a precise conjecture for the proportion of cubic surfaces in <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathbb {P}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> with a rational point.

How often does a cubic hypersurface have a rational point?

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2025年10月3日

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Lea Beneish,
Christopher Keyes

摘要

A cubic hypersurface in $\mathbb {P}^n$ $P^{n}$ defined over $\mathbb {Q}$ $Q$ is given by the vanishing locus of a cubic form f in $n+1$ $n + 1$ variables. It is conjectured that when $n \ge 4$ $n \geq 4$ , such cubic hypersurfaces satisfy the Hasse principle. This is now known to hold on average due to recent work of Browning, Le Boudec, and Sawin. Using this result, we determine the proportion of cubic hypersurfaces in $\mathbb {P}^n$ $P^{n}$ , ordered by the height of f, with a rational point for $n \ge 4$ $n \geq 4$ explicitly as a product over primes p of rational functions in p. In particular, this proportion is equal to 1 for cubic hypersurfaces in $\mathbb {P}^n$ $P^{n}$ for $n \ge 9$ $n \geq 9$ ; for $100\%$ $100 %$ of cubic hypersurfaces, this recovers a celebrated result of Heath-Brown that non-singular cubic forms in at least 10 variables have rational zeros. In the $n=3$ $n = 3$ case, we give a precise conjecture for the proportion of cubic surfaces in $\mathbb {P}^3$ $P^{3}$ with a rational point.