Uniform bounds for rational points on cubic hypersurfaces

Book chapter, 2015

© Cambridge University Press 2015. We use a global version of Heath-Brown’ p-adic determinant method to show that there are ON,ε(Bdim X + 1/7 + ε) rational points of height at most B on a geometrically integral variety X ⊂ PN of degree three defined over Q. By the same method we also show that there are Oε(B12/7 + ε) rational points of height at most B outside the lines on any cubic surface in P3.