Density of rational points on diagonal quartic surfaces
by
Ronald van Luijk
Warwick University
Coauthors: David McKinnon, Adam Logan

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.

Date received: April 30, 2008