Ronald van Luijk, University of Warwick, UK

Density of rational points on diagonal quartic surfaces

Abstract:
It is a wide open question whether the set of rational points
on a smooth quartic surface in projective three-space can be
nonempty, yet finite. In this talk I will treat the case of diagonal
quartics V, which are given by ax^4+by^4+cz^4+dw^4=0 for
some nonzero rational a,b,c,d. I will assume that the product
abcd is a square and that V contains at least one rational
point P. I will prove that if none of the coordinates of P is zero,
and P is not contained in one of the 48 lines on V, then the
set of rational points on V is dense. This is based on joint work
with Adam Logan and David McKinnon.