20240409 Chris Keyes

A cubic hypersurface in projective n-space defined over the rationals is given by the vanishing locus of an integral cubic form in n+1 variables. For n at least 4, it is conjectured that these varieties satisfy the Hasse principle. Recent work of Browning, Le Boudec, and Sawin shows that this conjecture holds on average, in the sense that the density of soluble cubic forms is equal to that of the everywhere locally soluble ones. But what do these densities actually look like? We give exact formulae in terms of the probability that a cubic hypersurface has p-adic points for each prime p. These local densities are explicit rational functions uniform in p, recovering a result of Bhargava, Cremona, and Fisher in the n=2 case, as well as the fact that all cubic forms are everywhere locally soluble when n is at least 9. Consequently, we compute numerical values (to high precision) for natural density of cubic forms with a rational point for n at least 4, and a conjectural value for n=3. This is joint work with Lea Beneish.