How often does a cubic hypersurface have a rational point?

• Lea Beneish ^[1] ; Christopher Keyes ^[2]
  1. [1] University of North Texas

     University of North Texas

     Estados Unidos

     
  2. [2] King's College London

     King's College London

     Reino Unido

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 5, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01079-w
• Enlaces
  • Texto completo
• Resumen

  • A cubic hypersurface in Pn defined over Q is given by the vanishing locus of a cubic form f in n +1 variables. It is conjectured that when n ≥ 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 Pn, ordered by the height of f , with a rational point for n ≥ 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 Pn for n ≥ 9; for 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 case, we give a precise conjecture for the proportion of cubic surfaces in P3 with a rational point.

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