Density of Rational Points on a Certain Smooth Bihomogeneous Threefold

Abstract

We establish sharp upper and lower bounds for the number of rational points of bounded anticanonical height on a smooth bihomogeneous threefold defined over Q and of bidegree (1, 2). These bounds are in agreement with Manin's conjecture.

The Manin–Peyre formula for a certain biprojective threefold

“…x 3 5 . Le Boudec [5] determined the order of magnitude for the number of rational points of bounded height on the biprojective threefold x 1 y 2 1 + x 2 y 2 2 + x 3 y 2 3 = 0, and we agree with him that a refinement to an asymptotic formula seems "far out of reach".…”

Linear Growth for Certain Elliptic Fibrations

“…This is achieved in Section 2.1. In the second step, we bound the number of nontrivial integral points of bounded height on the curves E n,e on average over n. To achieve this, we appeal to the recent result of the author [5,Lemma 4]. This lemma is stated in Section 2.2.…”

Rational points and non-anticanonical height functions

“…Le Boudec [LB15] has proved upper and lower bounds of the correct order of magnitude for the anticanonical height function for (1.2), but an asymptotic formula for the anticanonical height is still unknown in this case. Our methods also apply to cubic hypersurfaces.…”