Per Salberger

On the density of rational and integral points on algebraic varieties

Abstract Let X ⊂ ℙ n be a projective geometrically integral variety over of dimension r and degree d ≧ 4. Suppose that there are only finitely many (r − 1)-planes over on X. The main result of this paper is a proof of the fact that the number N(X;B) of rational points on X which have height at most B satisfies for any ɛ > 0. The implied constant depends at most on d, n and ɛ.

On a certain senary cubic form

A strong form of the Manin-Peyre conjecture with a power-saving error term is proved for a certain cubic fourfold.

Counting Rational Points On Threefolds

Let X ⊂ ℙ4 be an irreducible hypersurface and e > 0 be given. We show that there are O(B3+e), resp. O(B55/18+e), rational points on ℙ4 lying on X when X is of degree d ⩾ 4, resp. d = 3. The implied constants depend only on d and e.

Uniform bounds for rational points on cubic hypersurfaces

© Cambridge University Press 2015. We use a global version of Heath-Brown’ p-adic determinant method to show that there are ON,e(Bdim X + 1/7 + e) 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 Oe(B12/7 + e) rational points of height at most B outside the lines on any cubic surface in P3.