Sufficient conditions for the existence of rational points on diagonal cubic surfaces

Abstract

In this paper, we construct diagonal cubic surfaces over Q which have a Q-rational point under the assumption that the Tate-Shafarevich group of elliptic curve X3 + Y3 = AZ3 is finite. We can also check that there is no Brauer-Manin obstruction for these surfaces without the finiteness assumption of the Tate-Shafarevich group.