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SUMMARY:How often does a cubic hypersurface have a point? - Christopher Ke
 yes (King's College London)
DTSTART:20240220T143000Z
DTEND:20240220T153000Z
UID:TALK210550@talks.cam.ac.uk
CONTACT:Jef Laga
DESCRIPTION: A cubic hypersurface in projective n-space defined over the r
 ationals is given by the vanishing locus of an integral cubic form in n+1 
 variables. For n at least 3\, it is conjectured that the only obstruction 
 to rational points on cubic hypersurfaces are local ones --- that is\, the
 y satisfy the local-global principle. Recent work of Browning\, Le Boudec\
 , and Sawin shows that this conjecture holds on average for n at least 4\,
  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 ar
 e explicit rational functions uniform in p\, recovering a result of Bharga
 va\, Cremona\, and Fisher in the n=2 case\, as well as the fact that all c
 ubic forms are everywhere locally soluble when n is at least 9. Consequent
 ly\, we compute numerical values (to high precision) for natural density o
 f cubic forms with a rational point for n at least 4. This is joint work w
 ith Lea Beneish.\n
LOCATION:MR13
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